M»VB(sinr  OF 

DAVIS 


PHYSICAL    MEASUREMENTS 


IN  THE 


PROPERTIES   OF   MATTER 


AND  .IN 


HEAT 


BY 

RALPH    S.  MINOR 

Associate  Professor  of  Physics,  University  of  California 
AND 

T.  SIDNEY    ELSTON 

Instructor  in  Physics,  University  of  California 


BERKELEY,  CALIFORNIA 
1910 


OT 


Copyrighted  in  the  Year  1910  by 
RALPH  S.  MINOR  and  T.  SIDNEY  ELSTON 
In  the  Office  of  the  Librarian  of  Congress,  Washington 


PREFACE 

This  manual  is  printed  primarily  for  the  use  of  the  fresh- 
man students  in  the  various  engineering  colleges  of  the  Uni- 
versity of  California,  and  represents  the  laboratory  side  of  a 
three-unit  course  consisting  of  one  lecture,  one  recitation,  and 
one  laboratory  period  per  week  throughout  the  year.  The 
course  is  preceded  by  a  matriculation  course  in  Elementary 
Physics,  and  is  the  first  part  of  a  two-years  course  in  General 
Physics,  the  second  part  which  deals  with  Sound,  Light,  and 
Electricity  being  given  during  the  sophomore  year. 

The  predecessors  of  the  present  writers,  Professor  Harold 
Whiting,  Professor  Elmer  E.  Hall,  A.  C.  Alexander,  G.  K. 
Burgess,  Bruce  V.  Hill,  and  A.  S.  King,  have  all  played  a 
part  in  the  evolution  of  this  manual.  Its  present  form,  how- 
ever, is  largely  a  revision  of  the  manual  printed  in  1908  by 
Prof.  Hall  and  Dr.  Elston.  In  this  revision  the  present 
writers  have  been  guided  by  a  desire  to  so  simplify  the  work  as 
to  make  possible  the  performance  of  all  of  the  experiments, 
with  a  few  exceptions,  during  a  two-hour  laboratory  period. 
Free  use  has  been  made  of  published  texts  and  manuals  with- 
out specific  credit  being  always  indicated.  Several  experiments 
have  been  taken  from  the  sophomore  course  of  Professor  E. 
R.  Drew,  written  before  the  present  division  of  subjects  was 
made. 

RALPH  S.  MINOR, 
T.  SIDNEY  ELSTON. 

Berkeley,  Cal.,  August,  1910. 


LIST   OF    EXPERIMENTS 

1.  Sensitive  Beam  Balance.     Density  of  a  Solid. 

2.  Jolly's  Balance.     Specific  Gravity. 

3.  Model  Beam  Balance. 

4.  Boyle's  Law. 

5.  The  Volumenometer. 

6.  The  Force  Table. 

7.  Three  Forces  in  Equilibrium. 

8.  Density  of  Air. 

9.  Relative  Density  of  Carbon  Dioxide. 

10.  Uniformly  Accelerated  Motion. 

11.  Centripetal  Force. 

12.  The  Principle  of  Moments. 

13.  The  Simple  Pendulum. 

14.  The  Force  Equation. 

15.  Surface  Tension  by  Jolly's  Balance. 

1 6.  Capillarity.     Rise  of  Liquids  in  Tubes. 

17.  Rise  of  Liquids  between  Plates. 

1 8.  Viscosity.     Flow  of  Liquids  in  Tubes. 

19.  Efflux  of  Gases.     Relative  Densities. 

20.  Absolute  Calibration  of  a  Thermometer. 

21.  Relative  Calibration  of  a  Thermometer. 

22.  Variation  of  Boiling  Point  with  Pressure. 

23.  Coefficient  of  Expansion     of  a  Liquid     by     Archimedes' 

Principle. 

24.  Comparison  of  Alcohol  and  Water  Thermometers. 

25.  Coefficient  of  Expansion  of     a     Liquid     by     Regnault's 

Method. 

26.  Coefficient     of     Expansion     of     Glass  by  Weight-Ther- 

mometer. 

27.  Coefficient  of  Expansion  of  a  Liquid  by  Pycnometer. 

28.  Expansion  Curve  of  Water. 


LIST  OF  EXPERIMENTS  v 

29.  Specific  Heat  of  a  Liquid  by  Method  of  Heating. 

30.  Specific  Heat  of  a  Liquid  by  Method  of  Cooling. 

31.  Mechanical  Equivalent  of  Heat  by  Callendar's  Method. 

32.  Mechanical  Equivalent  of  Heat  by  Puluj's  Method. 

33.  Cooling  through  Change  of  State. 

34.  Heat  of  Fusion. 

35.  Heat  of  Vaporization  at  Boiling  Point. 

36.  Heat  of  Vaporization  at  Room  Temperature. 

37.  Freezing  Point  of  Solutions.     t 

38.  Heat  of  Solution. 

39.  Heat  of  Neutralization. 

40.  Expansion   of   Air   at     Constant     Pressure     by     Flask 

Method. 

41.  Expansion  of     Air     at     Constant     Pressure.    Constant- 

Pressure  Air- Thermometer. 

42.  Constant- Volume  Air-Thermometer. 

43.  Vapor-Pressure  and  Volume. 

44.  Vapor-Pressure  and  Temperature. 

45.  Hygrometry. 

46.  Density  of  the  Air  by  the  Barodeik. 

47.  Coefficient  of  Friction. 

48.  Conservation  of  Momentum.     Coefficient  of  Restitution. 

49.  Young's  Modulus  by  Stretching. 

50.  Hooke's  Law  for  Twisting.     Coefficient  of  Rigidity. 

51.  Friction  Brake.  Power  Supplied  by  a  Motor. 

52.  Absorption  and  Radiation. 

53.  Ratio  of  the  Two  Specific  Heats  of  a  Gas. 


REFERENCES 

The  following  list  of  books  includes  all  those  to  which 
reference  is  made  in  this  manual.  Some  of  them  are  text- 
books, some  laboratory  manuals,  and  other  books  of  tables.  A 
few  of  them  have  been  placed  on  the  window-shelf  for  gen- 
eral reference ;  the  others  can  be  drawn  at  the  desk. 

Text  Books- 
Duff  (Blakiston's)  :  Text-Book  of  Physics  (Second  Edi- 
tion. 

Edser :  Heat  for  Advanced  Students. 

Hastings  and  Beach :  General  Physics. 

Preston:  Theory  of  Heat  (Second  Edition). 

Watson:  Text-Book  of  Physics   (Fourth  Edition,  1903). 

Laboratory  Manuals. — 

Terry  and  Jones:  A  Manual  of  Practical  Physics,  Vol.  i. 
Millikan:  Mechanics,  Molecular  Physics  and  Heat. 
Watson  :  Text-Book  of  Practical  Physics. 

Books  of  Tables— 

Landolt  and  Bernstein:  Physical  and     Chemical  Tables. 
Smithsonian  Institute:  Physical  Tables. 
Whiting:   Physical  Tables. 


PHYSICAL   MEASUREMENTS. 


PROPERTIES   OF   MATTER,  AND   HEAT. 

This  book  is  intended  to  be  mainly  a  manual  of  directions. 
It  has  been  the  aim  of  the  authors  to  make  it  complete  enough^ 
however,  so  that  when  used  in  conjunction  with  >Blakiston's 
Physics,  the  class  text-book,  it  will  cover  the  minimum  re- 
quirement for  the  year's  work.  It  is  expected  that  the  student 
will  elect  to  consult  the  larger  reference-books  (a  liberal  num- 
ber of  copies  of  which  are  available  at  the  desk)  for  general 
notions  regarding  physical  measurements,  the  discussion  of  re- 
sults, the  effect  of  errors  in  observation  and  methods  for  their 
complete  or  partial  elimination. 

Printed  directions,  regarding  the  method  of  writing-up  and 
handing-in  the  record  of  the  experiments,  will  be  found  on 
the  folder  used  as  a  cover  for  the  record.  A  list  of  the  re- 
quired experiments  and  the  order  in  which  they  are  to  be  per- 
formed will  be  posted  on  the  laboratory  bulletin-board. 


1.     SENSITIVE  BEAM  BALANCE.     DENSITY  OF  A 

SOLID. 

Weighing  by  Method  of  Vibrations. 

In  weighing  with  a  sensitive  beam  balance  use  is  made  of  a 
long  pointer  attached  to  the  beam  and  arranged  to  vibrate  in 
front  of  a  fixed  scale.  The  more  sensitive  the  balance  the 
greater  the  angle  will  be  through  which  the  pointer  will  vi- 
brate for  a  given  excess  mass  placed  in  either  of  the  pans.  The 
pointer  will  swing  many  times  back  and  forth  before  it  finally 
comes  to  rest  at  a  definite  point  which  marks  the  position  of 


2  SENSITIVE  BEAM   BALANCE.    DENSITY  OF  A  SOLID.  [l 

equilibrium.  Time  would  be  wasted  in  waiting  for  it  to  stop, 
and  even  then  the  indications  of  the  moving  pointer  are  more 
trustworthy  than  those  of  one  which  has  come  to  rest,  because 
the  latter  may  not  be  in  its  true  position  of  equilibrium,  or  rest- 
point,  owing  to  friction. 

To  obtain  the  rest-point  the  pointer  is  allowed  to  vibrate  and 
the  turning-points  of  a  number  of  consecutive  swings  are  read, 
the  number  being  so  chosen  as  to  give  an  even  number  of 
turning-points  on  one  side  and  an  odd  number  on  the  other.  A 
little  consideration  will  show  that  under  this  condition  the 
point  halfway  between  the  mean  of  all  the  left-hand  and  the 
mean  of  all  the  right-hand  readings  is  the  true  rest-point. 
This  way  of  getting  the  rest-point  is  known  as  the  "method 
of  vibrations." 

To  weigh  a  body  it  is  necessary  first  to  know  the  rest-point 
with  the  two  pans  empty.  This  is  the  zero  rest-point.  The 
body,  being  placed  now  in  the  left-hand  pan,  enough  standard 
masses  are  placed  in  the  other  pan  to  balance  it.  If  the  rest- 
point  now  be  the  same  as  before,  the  weight  of  the  body  in  air 
is  represented  by  the  weight  of  the  standard  masses  used.  If 
the  rest-point  be  not  the  same,  it  is  best  then  to  determine  the 
sensitiveness  of  the  balance,  that  is,  the  number  of  scale-divi- 
sions through  which  the  addition  of  i  mg.  to  the  pan  will  shift 
the  rest-point.  From  this  and  the  difference  between  the  two 
rest-points,  the  weight  of  the  body  in  air  may  be  obtained  by 
calculation. 

In  most  sensitive  beam  balances  a  centigram  rider  is  used. 
By  properly  placing  the  rider  on  the  graduated  scale  attached 
to  the  beam,  the  equivalent  of  any  desired  mass  from  I  to  10 
mg.  may  be  added  to  either  pan.  Final  adjustments  can  thus 
be  made  without  opening  the  balance  case.  The  rider  should 
never  be  moved  without  first  lowering  the  balance  beam. 

The  balance  must  be  handled  with  the  greatest  care,  since 
any  jarring  or  rapid  vibration  of  the  beam  may  injure  the 


l]  SENSITIVE  BEAM  BALANCE.    DENSITY  OF  A  SOLID.  3 

knife-edges  upon  which  the  beam  rests.  On  this  account  the 
beam  should  be  lowered  each  time  before  a  mass  is  placed  on 
the  pan  or  removed  from  it,  and  also  when  the  weighing  is 
completed. 

To  illustrate  the  use  of  the  sensitive  beam  balance,  let  it  be 
required  to  find  the  density  of  a  cylindrical  solid. 

(a)  With  the  pans  of  the  balance  empty,    raise    the    beam 
slowly  and  allow  the  pointer  to  swing  over  four  or  five  scale- 
divisions.    Take  and  record  an  even  number  of  turning-points 
on  one  side  and  an  odd  number  on  the  other  (respectively  4 
and  3,  say),  and  from  them  determine  the  rest-point.     Make 
two  determinations  in  this  way,  and  take  the  mean  as  the  zero 
rest-point.    The  door  to  the  glass  case  should  always  be  closed 
when  determining  the  rest-point. 

(b)  Place  the  hard  rubber  cylinder  on  the  left-hand  pan 
of  the  balance,  and  add  masses  to  the  other    pan     until    the 
pointer  does  not  swing  off  the  scale  when  the  beam  is  raised. 
In  making  trials  for  the  correct  mass  on  the  right-hand  pan, 
raise  the  beam  only  high  enough  to  see  which  side     has    the 
greater  mass,  in  order  to  avoid  violent  rocking  of  the  beam. 
Use  the  fractional  masses  and  the  rider  to  bring  the  pointer  ap- 
proximately to  the  zero  rest-point,  and     then     determine    the 
rest-point  by  the  method  of  vibrations.  To  determine  how  much 
must  be  added  to,  or  subtracted  from,  the  masses  on  the  right- 
hand  pan  to  take  account  of  the  fact  that  the  rest-point  with 
the  loaded  balance  does  not  coincide  with  the  zero  or  empty- 
pan  rest-point,  by  means  of  the  rider  add  5  mg.  or  so  to  either 
pan   and  determine  the  sensitiveness   of  the  balance.     From 
the  sensitiveness,  the  difference  in  the     rest-points,     and     the 
masses  in  the  right-hand  pan,  find  the  exact  mass     which  will 
balance  the  hard  rubber  cylinder  in  air. 

(c)  Correction  for  Air-buoyancy. — Unless  the  body  whose 
mass  is  sought  has  the  same  density  as  the  masses     used     to 
balance  it,  the  body  will  be  buoyed  up  by  the  air  either  more 


4  JOLLY'S  BALANCE.    SPECIFIC  GRAVITY.  [2 

or  less  than  the  masses  are  buoyed  up,  and  this  will  introduce 
an  error  which  is  by  no  means  negligible  in  careful  measure- 
ments. To  correct  for  air-buoyancy :  Measure  the  dimensions 
of  the  cylinder  with  vernier  calipers,  and  compute  its  volume. 
Calculate  the  volume  of  the  standard  brass  masses  from  their 
marked  mass  values  and  the  density  of  brass  (8.4  gms.  per  cc.) 
Rea4  the  thermometer  and  barometer.  From  the  given  vol- 
umes and  the  density  of  air  at  the  temperature  and  barometric 
pressure  at  the  time  of  the  experiment  (see  Tables),  determine 
the  correction  for  air-buoyancy.  Calculate  the  density  of  the 
cylinder. 

(d)  If  the  arms  of  the  balance  be  unequal  in  length,  "double 
weighing"  is  necessary.  In  such  a  case  the  cylinder  is 
placed  in  the  right-hand  pan  and  the  mass  found  as  before. 
The  true  mass  is  then  given  by 

m  = 

where  m^  and  mz  are  the  values  obtained  by  the  two  weighings. 
The  proof  of  this  by  an  application  of  the  principle  of  mo- 
ments is  left  to  the  discretion  of  the  student.  In  most  sensi- 
tive balances  the  arms  are  so  nearly  equal  that  double  weigh- 
ing is  unnecessary. 

2.  JOLLY'S  BALANCE.    SPECIFIC  GRAVITY. 

Jolly's  balance  consists  of  a.  long  spiral  spring  suspended 
from  an  upright  steel  frame.  The  lower  end  of  the  spring  car- 
ries two  light  pans,  the  lower  of  which  is  always  immersed  to 
the  same  depth  in  a  beaker  of  water  standing  on  a  small  plat- 
form attached  to  the  frame.  In  one  form  of  the  balance  the 
upper  end  of  the  spring  is  stationary,  and  the  lower  end  car- 
rying the  pans  may  be  raised  or  lowered  and  the  elongation 
read  from  a  graduated  mirror  fixed  to  the  frame  just  behind 
the  spring.  In  the  other  form  of  the  balance  the  upper  end 
of  the  spring  is  movable  and  the  lower  end  with  the  two  pans 


2]  JOLLY'S  BALANCE.    SPECIFIC  GRAVITY.  5 

is  kept  at  a  fixed  mark;  the  elongation  is  read  from  a  grad- 
uated sliding  scale  attached  to  the  upper  end  of  the  spring. 

The  use  of  this  balance  to  determine  specific  gravity  depends 
on  the  fact  that  the  spring  obeys  Hooke's  law  closely  for  small 
elongations,  that  is,  the  elongation  is  proportional  to  the  change 
in  the  stretching  force.  With  the  lower  pan  immersed  in 
water  we  first  note  the  "empty-pan,"  or  zero,  scale-reading. 
The  solid,  whose  specific  gravity  is  to  be  found,  is  then  placed 
in  the  upper  pan,  elongating  the  spring  and  requiring  a  read- 
justment along  the  scale.  The  scale-reading  is  again  noted 
and  the  elongation  determined.  This  elongation  represents  the 
weight  of  the  solid  in  air.  The  solid  is  now  transferred  from 
the  upper  pan  to  the  lower  pan  and  the  elongation  of  the 
spring,  again  from  the  zero  position,  is  determined.  This 
latter  elongation  represents  the  weight  of  the  solid  in  water. 
From  these  data  the  elongation  equivalent  to  the  weight  of 
water  displaced  by  the  solid  may  be  found  and  the  specific 
gravity  of  the  solid  calculated. 

(a)  Hooke's  Law  Tested. —  Place  a  standard  5  eg.  mass  in 
the  upper  pan  and  determine  the     elongation.     Repeat     with 
larger  masses,  up  to  5  gm.  or  more.    With  one  form  of  the  bal- 
ance the  readings  are  taken  by  bringing  a  bead  or  point  of  wire 
at  the  lower  end  of  the  spring  into  coincidence  with  its  image 
in  the  mirror  scale ;  with  the  other  form  the  readings  are  taken 
from  the  vernier  and  sliding  scale  after  bringing  the  mark  at 
the  lower  end  of  the  spring  on  a  level  with  the  etched  line  on 
the  glass  tube.     From  the  observed  elongations  determine  if 
the  spring  obeys  Hooke's  law  or  not. 

(b)  Solids  Heavier  Than  Water. — By  the  method  outlined 
above,  find  the  weights,  in  air  and  in  water,  of  the  solids  fur- 
nished.   Take  care  that  the  lower  pan  is  immersed  to  the  same 
depth  in  the  water  and  that  no  air-bubbles  cling  to  the  lower 
pan  or  to  the  solid  when  immersed.     From  the  data  obtained 
calculate  th^  specific  gravity  of  each  of  the  solids. 


6  MODEL    BEAM    BALANCE.  [3 

(c)  Solids  Lighter  Than  Water. — Find  the  specific  grav- 
ity of  a  solid  which  floats  in  water.    For  this  purpose  a  sinker 
must  be  used,  but  it  may  be  left  in  the  lower  pan  throughout 
the  experiment.     The  following  readings  will  be  found  nec- 
essary: first  with  the  upper  pan  empty,  then  with  the  solid  in 
the  upper  pan,  and  again  with  the  solid  tied  to  or  wedged  un- 
der the  sinker  in  the  lower  pan. 

(d)  Liquids. — Find  the  specific  gravity  of  a  salt-solution 
by  using  a  solid  hung  by  a  thread  instead  of  using  the  lower 
pan.  (Salt-water  will  corrode  the  nickeled  surface  of  the  pan.) 
It  will  be  necessary  to  find  the  elongation  of  the  spring  equi- 
valent to  the  displacement  of  the  solid  in  water  as  well  as  in 
the  salt-solution.     Explain  the  method  used. 

(e)  How  would  you  proceed  to  use  Jolly's     balance     to 
weigh  objects  as  you  use  the  beam  balance? 

If  a  bubble  of  air  had  been  carried  down  with  the  solid 
when  immersed,  would  the  calculated  specific  gravity  have 
been  greater  or  less  than  it  should  be? 

Why  is  it  necessary  to  keep  the  lower  pan  immersed  always 
to  the  same  depth  throughout  a  given  experiment? 

What  additional  data  would  you  need,  if  required  to  calcu- 
late the  density  from  the  specific  gravity  of  any  of  the  objects 
used  in  this  experiment?  Explain. 

3.    MODEL  BEAM  BALANCE. 

The  beam  balance  consists  of  a  metal  beam,  supported  so  as 
to  be  able  to  rotate  about  a  central  knife-edge  located  verti- 
cally above  the  center  of  gravity  of  the  beam.  Near  the  ends 
of  this  beam,  pans  are  hung  from  knife-edges.  The  result  is 
that,  wherever  the  object  and  the  standard  masses  may  be 
placed  in  the  two  pans,  the  vertical  force  which  keeps  them  in 
equilibrium  must  pass  through  the  knife-edge  above,  and  so 
the  effect  upon  the  balance  is  the  same  as  if  the  whole  weight 


3] 


MODEL    BEAM    BALANCE. 


of  the  scale-pan  and  included  load  acted  at  some  point  in  the 
knife-edge  from  which  the  pan  is  hung.  The  distance  from 
the  central  knife-edge  to  the  knife-edge  at  either  end  of  the 
beam  is  called  the  arm  of  the  balance  or  the  length  of  the  beam. 
A  model  beam  balance  is  a  simplified  beam  balance  used  to 
test  the  relation  between  the  sensitiveness  of  the  balance  and 
its  dimensions  and  load.  By  "sensitiveness"  is  meant  the  fa- 
cility with  which  the  pointer  of  the  balance  can  be  deflected 
when  there  is  a  small  difference  between  the  masses  suspended 
from  the  two  sides  of  the  beam.  The  sensitiveness  of  the  bal- 
ance depends  upon  the  length  and  mass  of  the  beam,  the 
load  in  the  pans,  the  distance  between  the  center  of  gravity  of 
the  beam  and  the  central  supporting  knife-edge,  and  upon 
whether  the  beam  is  straight  or  curved  up  or  down.  To  obtain 
an  expression  showing  the  character  of  this  dependence  we 
need  to  apply  the  principle  of  moments. 


Fig.  i. 

Let  us  suppose  that  in  the  figure  the  points  A,  C,  B  rep- 
resent the  positions  of  the  three  knife-edges  and  G  the  posi- 
tion of  the  center  of  gravity  of  the  beam  when  the  two  pans 
are  carrying  equal  loads ;  and  that  the  points  A',  C,  B'  repre- 


8  MODEL    BEAM     BALANCE.  [3 

sent  the  corresponding  positions  of  the  knife-edges  and  G'  the 
corresponding  position  of  the  center  of  gravity  when  a  small 
excess  mass  is  added  to  the  right-hand  pan. 
L,et  m  =  the  mass  of  the  beam, 

/  =  the  length  of  the  beam-arm  (the  two    being    as- 
sumed equal), 
M  =  the  mass  hung  on  each    side,  including  the    mass 

of  the  scale-pan, 
h  =  the  distance  from  the  central  knife-edge     to     the 

center  of  gravity  of  the  beam, 
x  =  a  small  excess  mass  placed  in  one  pan, 
a  =  the  deflection  produced  by  the  addition  of  x, 
ft  =  the  angle,  for  the  given  load,  between  a  horizon- 
tal line  and  the  line  drawn  from  the  central  knife-edge  to  the 
knife-edge  at  either  end,  when  the  beam  is  so  placed  that  the 
two  angles  which  can  be  thus  formed  are  equal.     ft     will  be 
positive  if  the  beam  is  concave  upwards,  negative  if  the  beam 
is  concave  downwards.     Applying  the  principle  of  moments 
for  the  case  of  equilibrium,  the  central  knife-edge  being  the 
center  of  moments,  we  have 

(I)       (M   +  *)    gl  COS    (ft  —  a)    --  Mgl  COS    (ft   +  a) 

—  mgh  sin  a  =  o. 
Expanding,  collecting  terms,  and  transposing, 

[mh  —  (2M  -j-  JF)  sin  ft]  sin  a  =  Ix  cos  ft  cos  a,  or 

,  tan  « /'cos  /9 

x      =  mh  —  (2M  +  *)  /sin  ,3' 
If  the  beam  is  straight,  ft  =  o  and 

tan  «          / 
(3)  ~T    =mh 

The  sensitiveness  is  measured  by  the  ratio,  tan  a  /x.  Since 
the  expression  for  this  ratio  does  not  contain  M,  it  follows 
that  the  sensitiveness  in  the  case  of  a  straight  beam  is  inde- 


3]  MODEL    BEAM    BALANCE.  9 

pendent  of  the  load ;  it  increases  with  any  arrangement  which 
makes  the  fraction  l/mh  larger.  In  the  case  of  a  curved  beam, 
however,  it  is  evident  from  (2)  that  the  sensitiveness  is  de-: 
pendent  upon  the  load  and  also  upon  the  extent  and  direction 
of  the  curvature. 

The  model  balance  provided  allows  ample  modification  of 
the  several  quantities  in  equation  (2),  with  the  exception  of 
the  mass  m  of  the  beam.  The  following  possibilities  are  at 
once  apparent :  ( I )  the  length  /  of  the  beam  may  be  varied  by 
loosening  the  set  screws  which  clamp  the  movable  end-por- 
tions of  the  beam  to  the  tubular  central  portion;  (2)  its  cen- 
ter of  gravity  may  be  raised  or  lowered  (thus  altering  h)  by 
sliding  the  metal  bob  up  or  down  along  the  pointer;  (3)  the 
load  M  may  be  increased  by  adding  masses  to  the  scale-pans  j 
(4)  the  points  of  suspension  of  the  two  pans  may  be  placed 
level  with,  above,  or  below  the  central  knife-edge,  thus  mak- 
ing the  beam  straight,  or  curved  up  or  down. 

Straight  Beam. 

(a)  Adjust  the  sliding  bob  so  that  the  center  of  gravity  of 
the  beam  lies  below  the  central  knife-edge.     Change  the  set 
screws  on  the  beams,  if  necessary,  so  as  to  make  the  two  beam- 
arms  equal  in  length.     Place  the  terminal  knife-edges  at  the 
zero  mark,  thus  insuring  a  straight  beam.     Hang     the     two 
scale-pans  in  position.     Level  up  the  balance  by  means  of  the 
thumb-screws  on  the  legs. 

Place  successively  several  2  eg.  masses  in  the  right-hand 
pan,  recording  the  deflections  and  noting  if  they  are  propor- 
tional to  the  number  of  masses  used.  Why  should  the  de- 
flection not  be  strictly  proportional  to  the  number  of  masses? 

For  convenience,  select  the  deflection  produced  by  the  first 
2  eg.  mass  as  a  measure  of  the  sensitiveness  of  the  balance. 

(b)  Change  the  length  of  the  beam-arm.     Test  the  sensi- 
tiveness by  means  of  a  2  eg.  mass.     Compare  with  (a),  and 


IO  MODEL   BEAM    BALANCE.  f3 

state  how  the  sensitiveness  depends  upon  the  length  of  the 
beam-arm,  other  conditions  remaining  unchanged;  and  note 
if  the  result  is  in  agreement  with  the  formula. 

(c)  Adjust  the  length  of  the  beam-arm  back  to  its  value  in 
(a),  and  then  change  the  position  of  the  center  of  gravity  of 
the  beam  by  sliding  the  bob  up  or  down.     Test  the  sensitive- 
ness and  compare  with   (a),  giving  your  conclusions. 

(d)  Bring  the  bob  back  again  to  its  position  in  (a).     In- 
crease the  load  by  placing  a  100  gm.  mass  in  each  scale-pan. 
Test  the  sensitiveness  and  compare  with  (a),  giving  your  con- 
clusions. 

Curved  Beam. 

(e)  Remove  the  masses  from  the  scale-pans.     Raise     the 
terminal  knife-edges  to  the  first  mark  above  zero,  thus  chang 
ing  the  beam  from  a  straight  beam  into  one  which  is  curved 
up.     Test  the  sensitiveness  and  compare  with  (a).     Does  in- 
creasing the  upward  curvature  of  the  beam,  other  conditions 
remaining  unchanged,  increase  or  decrease    the    sensitiveness  ? 
Show  that  this  is  in  agreement  with  the  formula. 

(f)  Place  a  100  gm.  mass  in  each  scale-pan.    Test  the  sen- 
sitiveness and  compare  with  (e),  giving  your  conclusions. 

(g)  Remove  the  masses  from  the  scale-pans.     Lower  the 
terminal  knife-edges  to  the  first  mark  below  the     zero,     thus 
changing  the  beam  into  one  which  is  curved  down.     Test  the 
sensitiveness  and  compare  with  (a),  giving  your  conclusions. 

(h)  Place  a  100  gm.  mass  in  each  scale-pan.  Test  the  sen- 
sitiveness and  compare  with  (g),  giving  your  conclusions. 

A  sensitive  chemical  balance  is  usually  made  with  the  beam 
curved  slightly  upwards  when  there  is  no  load  in  the  pans.  A 
medium  load  straightens  the  beam  and  an  excess  load  causes 
a  downward  curvature.  From  the  results  above,  state  how 
the  sensitiveness  of  such  a  balance  will  change  with  the  load 
on  account  of  the  curvature  of  the  beam. 


4]  BOYLE'S  LAW.  n 

4.     BOYLE'S  LAW. 

Reference.— Duff,  p.  158. 

The  purpose  of  this  experiment  is  to  study  the  relation  be- 
tween the  volume  and  the  pressure  of  a  given  mass  of  air  kept 
at  constant  temperature.  According  to  Boyle's  law  the  vol- 
ume of  a  fixed  mass  of  gas,  kept  at  constant  temperature,  var- 
ies inversely  as  the  pressure  in  the  gas.  This  relation  may 
be  mathematically  expressed  in  various  ways :  ( I )  The  vol- 
umes of  the  gas  at  two  different  times  are  inversely  propor- 
tional to  the  corresponding  pressures;  (2)  The  product  of  the 
volume  and  pressure  at  one  time  is  equal  to  the  corresponding 
product  at  another  time,  or,  in  other  words,  the  product  of  the 
volume  and  pressure  of  the  gas  is  a  constant,  that  is, 

*V— it, 

where  p  and  v  are  respectively  the  pressure  and  corresponding 
volume,  and  K  is  a  constant  whose  value  is  fixed  so  long  as 
the  temperature,  mass,  and  nature  of  the  gas  remain  un- 
changed. This  is  approximately  true  for  most  of  the  perma- 
nent gases,  provided  the  pressures  are  not  very  large.  The 
higher  the  temperature  at  which  the  gas  is  held,  the  more 
closely  does  the  law  hold  true. 

A  given  mass  of  dry  air  is  enclosed  in  an  inverted,  grad- 
uated glass  tube  which  is  attached  to  one  end  of  a  rubber  tube 
containing  mercury.  At  the  opposite  end  of  the  rubber  tube 
is  an  open  glass  tube.  The  glass  tubes  are  clamped  to  verti- 
cal guides  having  a  meter  scale  between  them.  The  two 
clamps  can  be  so  adjusted  along  the  guides  as  to  vary  the 
pressure  on  the  enclosed  air  from  values  greater  than  atmos- 
pheric pressure  to  those  which  are  smaller.  By  reading  the 
positions  of  the  menisci  of  the  mercury  columns  in  the  two 
tubes  and  adding  to,  or  subtracting  from,  the  atmospheric 


12  BOYLE'S  LAW.  [4 

pressure  as  determined  by  the  barometer,  the  pressure  corres- 
ponding to  each  setting  can  be  found.  The  corresponding 
volume  of  the  enclosed  air  can  be  read  from  the  graduated 
tube  containing  it. 

(a)  Clamp  the  tube,  enclosing  the  air,  to  the  vertical  guide 
near  the  bottom.  Raise  the  open  tube  as  high  along  the  other 
vertical  guide  as  possible  and  clamp  it  to  the  guide.  It  may 
be  necessary  to  pour  more  mercury  into  the  open  tube  so  as 
to  raise  the  mercury  level  on  that  side.  Record  the  temper- 
ature, the  volume  of  the  enclosed  air,  and  the  positions  of  the 
two  mercury  menisci.  Care  should  be  taken  in  reading  the 
position  of  the  mercury  meniscus  to  avoid  parallax.  For  this 
purpose,  in  sighting,  stand  so  that  the  eye,  the  mercury  sur- 
face and  the  image  of  the  mercury  surface  formed  by  the 
mirror  are  in  the  same  straight  line.  In  reading  the  volume 
of  the  enclosed  air,  account  should  be  taken  of  the  curvature 
of  the  mercury  meniscus.  Read  the  barometer  and  record 
the  atmospheric  pressure  in  cm.  of  mercury. 

(£>)  Lower  the  open  tube  10  cm.  or  so  at  a  time,  repeating 
the  readings  in  (a)  for  each  position.  Continue  until  the  two 
mercury  surfaces  are  in  the  same  level.  Record,  in  tabular 
form,  the  meniscus-readings^  the  volume  v,  the  total  pressure 
/>,  and  the  product  p  v. 

(c)  Unclamp  the  two  tubes  and  raise  them  to  points  near 
the  upper  ends  of  the  vertical  guides.    In  doing  so,  care  should 
be  taken  not  to  allow  mercury  to  overflow.     Lower  the  open 
tube  10  cm.  or  so  at  a  time,  repeating  the  readings  and  tabu- 
lating as  in   (b). 

(d)  On  separate  sheets  of  millimeter  cross-section  paper 
plot  the  following  curves : 

(i)  With  pressures  as  ordinates  and  volumes  as  abscissae, 
plot  the  data  of  (a),  (&),  and  (c).  Draw  a  smooth  curve 
which  will  best  represent  the  average  position  of  the  plotted 
points.  What  mathematical  curve  does  it  resemble?  The  re- 


5] 


THE  VOLUMENOMETER. 


semblance  would  be  much  more  marked,  if  the  same  length  had 
been  chosen  to  represent  the  pressure-unit  and  the  volume- 
unit.  This,  however,  it  will  not  be  found  convenient  to  do 

(2)  With  the  products  p  v  as  ordinates  and  the  volumes 
as  abscissae,  plot  the  results  of  (a),  (&),  and  (c).  Draw  a 
smooth  curve  which  will  best  represent  the  plotted  points. 
What  mathematical  curve  does  it  most  closely  resemble? 
If 'Boyle's  law  held  strictly,  what  form  should  the  curve  take? 

(e)  The  temperature  prevailing  during  the  experiment  was 
room-temperature.  If,  instead,  a  higher  temperature  had  pre- 
vailed, state  with  your  reasons  how  you  should  expect  the 
curves  to  be  displaced  in  the  plots.  What  effect  would  an  in- 
crease in  the  mass  of  the  air  have,  other  conditions  remaining 
the  same  as  in  the  experiment? 

What  are  the  principal  sources  of  error  ? 

Determine  from  the  two  plots  the  volumes  of  the  enclosed 
air  for  pressures  of  50  and  100  cm.  Compare. 

5.    THE  VOLUMENOMETER. 

The  object  of  this  experiment  is  to  find  the  density  of  an  ir- 
regular solid  by  means  of  the  volumenometer  and  the  balance. 
In  the  volumenometer,  A  is  a  glass  tube  which  may  be  closed 


Fig.  2. 


i.f 


14  THE   VOLUMENOMETER.  [5 

at  the  top  by  a  ground  glass  plate.  It  corresponds  to  the  closed 
tube  in  the  experiment  on  Boyle's  law.  As  in  that  experiment, 
the  pressure  and  volume  of  the  air  in  A  are  varied  by  raising 
or  lowering  a  tube  containing  mercury.  The  pressure  is  de- 
termined by  noting  the  difference  in  the  levels  of  the  two  mer- 
cury menisci,  and  adding  to  or  subtracting  from  the  atmos- 
pheric pressure  as  read  from  the  barometer.  The  volume  is 
unknown.  The  volume  of  a  portion  of  the  tube  between  two 
marks  (M  and  N),  however,  is  known. 

Let  the  volume  between  M  and  N  be  k,  and  that  above  M  be 
V.  By  determining  the  pressures  when  the  volume  of  air  is 
V  (mercury  meniscus  at  M),  and  again  when  the  volume  is 
V-\-k  (mercury  meniscus  at  N),  an  equation  involving  Boyle's 
law  may  be  written  containing  these  two  volumes  and  the  cor- 
responding pressures.  From  this  equation  V  may  be  calcu- 
lated. The  volume  of  air  in  A  may  be  found  in  this  way 
both  with  and  without  the  solid  body  enclosed  whose  volume 
we  desire  to  know.  The  volume  of  this  solid  thus  becomes 
known.  From  its  volume  and  mass  its  density  can  be  found. 

(a)  With  the  tube  A  uncovered  bring  the  mercury  meniscus 
to  M,  recording  the  pressure,  evidently  just  equal  to  the  atmos- 
pheric pressure.     Carefully  place  the  plate  on  A,  so  as  to  in- 
sure an  air-tight  joint.     The  plate  must  be  clean  and  have  on 
it  only  a  little  grease.     Lower  the  mercury  meniscus  in  the 
tube  A  from  M  to  N,  note  the  difference  in  level  of  mercury 
menisci  in  A  and  L,  and  again  determine  the  pressure.     Test 
for  leakage  by  allowing  the  tube  to  remain  a  minute  or  more 
in  this  position,  and  make  sure  that  the  heights  of  the  menisci 
do  not  change.     Calculate  k,  and  then,  by  applying     Boyle's 
law,  find  V. 

(b)  Remove  the  plate,  place  inside  the  volumenometer  one 
of  the  bodies  whose  density  is  to  be  determined,  and  repeat 
(a).     From  the  volume  V  of  the  air,  found  in  this  case,  and 


6]  THE  FORCE  TABLE.  15 

the  former  volume  V,  the  volume  of  the  body  is  found.  Weigh 
the  body  and  determine  its  density. 

(c)  Repeat  for  at  least  two  other  bodies. 

(d)  If  it  were  possible  to  perfect    the  measuring    instru- 
ments used  in  the  course  of  this  experiment  so  that  they  would 
be  absolutely  accurate,  it  would  still  be  unreasonable  to  expect 
that  the  results  obtained  for  the  density  would  be  those  given 
in  the  Tables.    Why  ? 

Determine  the  precision  of  measurement  of  the  balance,  the 
barometer,  and  the  volumenometer,  and  from  these  determine 
the  precision  of  measurement  or  reliability  of  the  final  result. 

What  are  the  advantages  and  disadvantages  of  this  method 
of  determining  density? 


6.    THE  FORCE  TABLE. 

Reference.— Duff,  pp.  36-41. 

The  purpose  of  this  experiment  is  to  determine  the  vector 
sum  or  resultant  of  two  forces  acting  on  a  body  in  the  same 
plane  and  along  lines  not  parallel.  Let  the  lines  of  direction 
of  the  two  forces,  /x  and  /2,  intersect  in  a  point,  O,  making 
angles,  a±  and  a2,  with  an  arbitrarily  chosen  axis,  OX.  The 
vector  sum  of  these  forces  is  by  definition  a  force,  f,  given  bv 
the  diagonal  of  the  parallelogram  formed  by  /\  and  fz  as 
sides.  Let  the  directional  angle  of  f  be  a.  By  taking 
the  projections  of  flt  f2,  and  /  upon  the  perpendicular  axes, 
OX  and  OF,  we  see  by  construction  that 

(1)  /  sin  a  =  A   sin  a:  +   f2  sin  a2, 

(2)  /  cos  a  =  /±  cos  ai  +  f2  cos  a2 ; 
whence,  by  squaring  (i)   and   (2)   and  adding, 

(3)  f2  =  A2  +  /22  +  2fJ,  c 


16  THE  FORCE  TABLE.  [6 

and,  by  dividing  (i)  by  (2), 

(4)         tana=/'sina'+/'sing'. 

/,  COS  a,  -f  /2  COS  oa 

The  last  two  equations  give  the  magnitude  and  direction 
of  the  diagonal  of  the  parallelogram  in  terms  of  the  magni- 
tudes and  directions  of  the  sides.  Equation  (3)  enables  one 
to  calculate  the  magnitude  of  the  resultant  of  the  two  given 
forces,  and  by  equation  (4)  the  direction  of  this  resultant  can 
be  determined. 

The  apparatus  used  to  test  these  results  consists  of  an  ad- 
justable iron  table  with  circular  top  graduated  in  degrees. 
Pulleys  can  be  clamped  to  the  circumference  at  any  chosen 
points.  From  a  pin,  placed  in  a  hole  in  the  center  of  the  table - 
top,  three  cords  pass  over  the  pulleys  and  carry  pans  upon 
which  known  masses  are  placed.  The  masses  and  pans  should 
be  weighed  on  the  platform  scales. 

Tested  Algebraically. 

(a)  Arbitrarily  take  flf  f2,  a±,  a2  as  equal  respectively  to 
(200  +  m)  gms.  wt.,  (100  +  ' m)  gms.  wt,  35°,  85°,  where 
m  is  the  mass  of  the  pan  holding  the  masses.  Calculate  by 
equations  (3)  and  (4)  the  value  of  /  and  a.  It  will  be  found 
advisable  to  make  these  calculations  and  those  in  (&)  before 
entering  the  laboratory  to  begin  the  experiment. 

Set  one  of  the  pulleys  at  35°  and  one  at  85°.  With  the  pin 
in  place,  put  the  requisite  masses  in  the  pans  to  make  /^  and 
/2  equal  to  the  values  chosen  for  them.  Then,  if  a  third  pul- 
ley be  set  1 80°  from  the  direction  determined  by  a  as  calcu- 
lated above,  and  masses  corresponding  to  the  calculated  value 
of  /  be  added,  the  three  forces  acting  on  the  pin  should 
be  in  equilibrium,  since  the  third  force  is  equal  and  opposite 
to  the  vector  sum  of  fl  and  f2.  Pull  out  the  pin  and  see  wheth- 
er the  calculation  is  correct. 


7]  THREE)  FORCES  IN  EQUILIBRIUM.  17 

(&)  In  a  similar  manner  calculate  and  test  two  other  sets 
of  values  chosen  by  you. 

Tested  Geometrically. 

(c)  Select  three  new  sets  of  values  for  flf  f2,  aif  and  a2  and 
proceed  as  follows  with  each  set:     Place  a  circular  sheet  of 
manila  paper  on  the  table  and  run  the  pin  through  it.     Set 
the  two  pulleys  at  at  and  a2,  and  place  the  requisite  masses 
on  the  pans.     Mark  with  a  pencil  the  directions  of  the  two 
strings,  then  remove  the  paper  and  on  the  lines  lay  off  dis- 
tances from  their  intersection  proportional  to  f1  and  /2.     Com- 
plete the  parallelogram  and  determine  from  the  diagonal  the 
value  of  /.     Now  replace  the  paper  on  the  table,  set  the  third 
pulley  opposite  to  /,  and  adjust  the  masses  on  its  pan  to  equal 
the  value  of  /  as  determined  by  the  diagonal.     Pull  out  the 
pin  and  see  if  the  construction  is  correct. 

(d)  Point  out  the  principal  sources  of  error     in  the    two 
methods  used  above. 

If  three  forces  acting  upon  a  body  hold  it  in  equilibrium, 
how  must  their  lines  of  direction  intersect? 

A  ladder  leaning  against  a  smooth  vertical  wall  is  prevented 
from  sliding  by  the  reaction  of  the  ground.  What  forces  are 
acting  on  the  ladder?  Construct  the  line  of  direction  of  the 
reaction  of  the  ground  on  the  ladder. 

7.     THREE  FORCES  IN  EQUILIBRIUM. 

Reference.— Duff,  p.  78. 

The  purpose  of  this  experiment  is  to  study  the  conditions 
which  must  be  satisfied  in  order  to  produce  equilibrium 
among  three  forces,  two  of  which  are  mutually  perpendicular. 
The  simplest  case  is  where  the  three  forces  are  applied  at  the 
same  point  in  the  body;  but  the  more  general  case  where  the 
three  forces  have  different  points  of  application  in  the  body  is 


i8 


THREE  FORCES  IN   EQUILIBRIUM. 


[7 


essentially  the  same,  in-so-far  as  equilibrium  is  concerned, 
for  the  lines  of  direction  of  the  three  forces  must  pass  through 
one  and  the  same  point  if  the  forces  are  in  equilibrium.  It  is 
quite  evident  that  the  three  forces  must  lie  in  the  same  plane, 
for  each  of  the  three  must  be  opposite  to  and  in  the  same 
straight  line  with  the  resultant  of  the  other  two.  Moreover, 
the  resultant  or  vector  sum  of  the  three  forces  must  be  zero, 
if  the  forces  are  in  equilibrium. 

Let  OA,  OB,  OC,  (see  Fig.  3)  represent  three  forces  flt  f2, 


/3,  which  are  in  equilibrium,  the  first  two  being  mutually  per- 
pendicular to  each  other.  The  third  force  /3,  to  produce  equi- 
librium, must  be  equal  and  opposite  to  and  in  the  same 
straight  line  with  the  resultant  of  the  other  two.  Since  the 
diagonal  OC  represents  the  resultant  of  f^  and  f2,  it  follows 
that  the  line  OC  which  represents  fs  must  be  equal  and  oppo- 
site to  and  in  the  same  straight  line  with  OC.  This  is  the 
point  of  view  considered  and  verified  by  Exp.  6. 

Since  the  forces  are  in  equilibrium,  their  resultant  effect  in 
any  direction  must  be  zero,  that  is,  the  algebraic  sum  of  the 
projections  of  the  forces  in  that  direction  must  be  zero.  In 
•the  line  DA  the  effect  of  f1  is  f1  cos  o°,  the  effect  of  /2  is 


7] 


THREE  FORCES  IN   EQUILIBRIUM, 


/2  cos  90°,  and  the  effect  of  /3  is  /3  cos  a.  But  the  resultant 
effect  along  DA  is  zero,  hence 

/!  COS  0°    +  /2  COS  90°    +  /3  COS  a  =  O. 

Since  cos  o°  =  i  and  cos  90°  =  o,  the  equation  becomes 

(1)  /,  +  /3  COSa  =  0. 

Similarly,  the  resultant  effect  along  EB  must  be  zero,  and 
hence  the  sum  of  the  projections  of  the  forces  into  that  line 
must  be  zero,  or 

/!  sin  o°  +  f2  sin  90°  +  fs  sin  a  =  o. 
Since  sin  o°  =  o  and  sin  90°  =  i,  this  equation  becomes 

(2)  /2  +  /8  sin  a  =  o. 

(It  should  be  noted  that  sin  a  and  cos  a  are  negative  for 
an  angle  in  the  third  quadrant.)  Interpreted  geometrically, 
equation  ( i )  shows  that  OA  and  OD  are  equal  and  opposite ; 
equation  (2)  shows  that  OB  and  OH  are  equal  and  opposite. 

The  present  experiment  is  intended  to  verify  the  relations 
(i)  and  (2).  The  apparatus  consists  essentially  of  a  spring 
balance  P  (see  Fig.  4),  a  compression  spring  balance  Q,  and 


Fig.  4. 


2O  THREE    FORCES   IN    EQUILIBRIUM.  [7 

a  weight-hanger  R.  The  two  balances  are  attached  to  a  ver- 
tical rod  along  which  they  can  be  adjusted.  The  balance  P 
is  kept  in  a  horizontal  position  so  as  to  be  constantly  at  right 
angles  to  the  vertical  force  exerted  by  the  weight-hanger.  The 
three  forces  acting  at  O  are  respectively  a  horizontal  pull  to- 
ward the  right  by  the  balance  P,  an  oblique  thrust  upward  to 
the  left  by  the  rod  belonging  to  the  balance  Q,  and  a  vertical 
pull  downward  due  to  the  weight  of  the  hanger  R.  A  differ- 
ent arrangement  of  forces  is  obtained  by  changing  the  number 
of  masses  carried  by  R,  and  by  changing  the  direction  of  Q. 
On  a  sheet  of  paper  fastened  behind  the  balance  the  lines  of  di- 
rection of  the  three  forces  can  be  traced. 

(a)  Arrange  the  balances  and  the  weight-hanger  in  the 
manner  indicated  in  the  figure.     Hang  a  mass  of  9  kg.  on  the 
hanger,  and  adjust  the  balance  P  along  the  rod  until  the  an- 
gle MOR,  as  tested  by  means  of  a  square,     is  a  right    angle. 
Support  the  weight  of  the     two  balances  by     means  of  the 
hands ;  note  if  the  balance-readings   are  materially   changed, 
and  if  they  are,  ask  for  assistance  in  correcting  for  the  same. 
Record  the  readings  of  the  two  balances  and  the  total  weight 
of  the  hanger  and  masses.     On  a  sheet  of  paper,  held  behind 
the  balances,  make  a  trace  of  the  lines  of  direction  of  the 
three  forces  and  determine  the  angle   NOR.     The  sine  and 
cosine  of  this  angle  may  be  found   from  the  Tables,  or  di- 
rectly from  a  measurement  of  the  distances  OM  and  MN. 

(b)  Repeat  (a)  twice  with  different  masses  on  the  hanger. 
Readjust  the  balance  P  each  time  so  that  the  angle     MOR 
shall  equal  a  right  angle. 

(c)  Change  the  length  of  the  cord  or  chain  which  con- 
nects O  with  the  balance  P,  and  thus  alter  the  angle    NOR. 
Repeat  the  adjustments  and  measurements  of  (a). 

(d)  From   the   results   in    (a),    (&),   and    (c)    determine 
if  the  condition  of  equilibrium  is  satisfied,  first  by  substitut- 
ing the  recorded  values  in  the  equations    (i)   and   (2),  and 


8]  DENSITY   OF    AIR.  21 

again  by  constructing  the  triangle  of  forces  in  each  case  and 
noting  if  it  is  closed  or  not. 

(e)  On  one  of  the  traces  draw  through  O  a  line  which 
does  not  coincide  with  any  one  of  the  three  forces.  Deter- 
mine the  effect  which  each  one  of  the  three  has  in  this  line 
and  see  if  the  resultant  effect  is  zero,  employing  the  method 
already  outlined  in  deriving  equations  (i)  and  (2). 

What  are  the  principal  sources  of  error  in  this  experiment? 

How  can  equations  (i)  and  (2)  be  used  to  calculate  the 
magnitude  and  direction  of  the  resultant  of  forces  f±  and  /2? 

What  is  the  form  of  the  equations  which  would  connect 
/!,  /2,  and  /3  if  the  angle  NOR  were  not  a  right  angle  (see 
Exp.  6)  ? 

8.     DENSITY  OF  AIR. 

Let  a  glass  bulb  of  volume  V  be  weighed  full  of  air  at  at- 
mospheric pressure  P±,  and  let  M  be  the  mass  necessary  to 
balance  it.  Then  let  the  air  be  pumped  out  until  the  pressure 
is  P2,  the  mass  as  determined  by  weighing  now  being  (M  —  w), 
where  m  is  the  mass  of  air  that  has  been  pumped  out  between 
the  weighings.  Then  if  d±  and  d2  be  the  densities  correspond- 
ing to  the  pressures  Px  and  P2,  it  follows,  from  the  definition 
of  density  and  the  interpretation  of  m,  that 

(!)         Vdi—  Vd2  =  m. 

The  reciprocal  of  the  density  is  the  volume  of  unit  mass: 
hence,  if  Boyle's  law  is  applied  to  unit  mass  of  the  air,  the 
temperature  being  assumed  constant,  we  have  (see  Exp.  4), 


Eliminating  dz  from  (i)  and  (2),  we  get 
(3)        4 


</>-/>) 


22  DENSITY    OF    AIR.  [8 

This  last  equation  may  be  employed  as  a  formula  to  find  the 
density  of  the  air  from  a  knowledge  of  the  volume  of  the 
flask,  the  pressure  before  and  after  exhaustion,  and  the  mass 
of  air  pumped  out.  In  the  application  it  is  essential  that  the 
temperature  should  be  the  same  during  the  two  weighings. 
(Why?)  This  condition  is  approximately  satisfied  in  practice. 
If  the  temperature  were  not  the  same,  the  observed  pressure 
in  the  second  case  would  need  to  be  corrected  (through  the 
application  of  Charles'  law)  so  as  to  give  the  pressure  that 
would  have  existed  had  the  temperature  been  the  same  as  dur- 
ing the  first  weighing.  The  volume  V  is  obtained  by  weighing 
the  bulb  when  empty  and  then  when  full  of  water  at  a  known 
temperature. 

(a)  Carefully  dry  the  flask  by  exhausting  it  several  times 
and  admitting  air  each  time  through  a  calcium-chloride  drying- 
tube.     Ask  an  assistant  for  instructions  in  regard  to  manipu- 
lating the  pump.    If  moisture  is  visible  inside  the  flask,  it  may 
be  necessary  to  put  in  a  little  alcohol,  rinse  the  flask,  vaporize 
the  alcohol  over  a  Bunsen  burner,  and  rinse  with  dry  air  as  be- 
fore.   With  the  dried  flask  in  connection  with  the  drying  tube, 
admit  air  at  atmospheric  pressure.     Close  the  stop-cock  and 
carefully  weigh  the  flask.     Note  the  temperature.     Read  the 
barometer  for  the  pressure. 

(b)  Pump  the  air  out  until  as  low  a  pressure  as  possible 
is  obtained  and  weigh    again  at  this  reduced  pressure.  Again 
note  the  temperature  and  record  the  pressure. 

(c)  Fill  the  flask  completely  with  water  up  to  the     stop- 
cock, taking  care  to  have  no  water  above  it.    Ask  an  assistant 
to  show  you  how  to  fill  it.     The  temperature  of  the  water 
should  be  recorded  and  its  density     found  from  a     book  of 
Tables.    Dry  the  outside  of  the  flask  and  then  weigh.     Calcu- 
late the  volume  of  the  flask. 

(d)  Using  the  results  obtained  in  (a),  (b),  and  (c),  find 
the  density  of  the  air,  in  grams  per  cc.,  at  the  given  tempera- 


9]  RELATIVE    DENSITY    OF    CARBON    DIOXIDE.  23 

ture  and  atmospheric  pressure.  From  this  result  the  density 
or  dry  air  under  standard  conditions  (that  is,  at  o°C  and  760 
mm.  pressure),  may  be  found  through  the  application  of 
Boyle's  and  Charles'  laws,  or  a  combination  of  the  two.  If 
Plt  d^  and  T1  represent  the  pressure,  density  and  absolute 
temperature  of  a  given  kind  of  gas  at  one  time,  and  P2,  d2  and 
T2  represent  the  corresponding  values  at  another  time,  and  so 
on,  then  it  follows  from  a  combination  of  the  two  laws  that 

P  P 

(~\  -*!      -*» 

v3;        j  T  —  -TT 
ai2  i      aiL  i 

for  the  given  kind  of  gas  to  the  degree  of  approximation  with 
which  it  observes  the  given  laws.  Making  use  of  this  relation, 
calculate  the  density  of  dry  air  under  standard  conditions  of 
temperature  and  pressure,  and  compare  with  the  value  given 
in  the  Tables.  Point  out  the  chief  sources  of  error  and  any 
other  reasons  for  the  discrepancy  in  the  results. 


9.  RELATIVE  DENSITY  OF  CARBON  DIOXIDE. 

The  relative  density  of  carbon  dioxide  compared  with  air 
as  a  standard  is  to  be  measured.  The  method  employed  is  that 
used  in  Exp.  8.  Using  the  same  symbols  as  there  used,  and 
making  the  weighings  and  noting  the  pressures  as  there  indi- 
cated, we  have  for  the  air, 

(!)  rf,  = 

If  the  measurements  are  then  repeated  for  the  carbon  dioxide, 

(2)  d\  = 


the  symbols  having  the  same  meaning  as  in  the  case  of  air. 
From  (i)  and  (2),  if  D  is  the  relative  density  of  the  carbon 
dioxide,  we  get,  by  division, 


24  UNIFORMLY  ACCELERATED   MOTION.  flO 

4'  _  *^£-_^)  . 

-  4  -  «/>  (/Y  -  /Y) ' 

from  which  we  see  that  a  determination  of  the  volume  of  the 
flask  is  unnecessary. 

(a)  Read  the  directions  given  under  Exp.  8.  Ask  an  assist- 
ant for  instructions  in  the  use  of  the  pump.    Carefully  dry  the 
flask,  and  fill  it  with  dry  air  admitted  through  the  calcium 
chloride  tube.     Using  a  sensitive  balance,  weigh  the  flask  full 
of  air  at  atmospheric  pressure,  noting  the  pressure  and  temper- 
ature.    In  weighing,  follow  the  method  given  in  Exp.  I. 

(b)  Pump  the  air  out  until  a  low  pressure  is  obtained  and 
weigh  the  flask  again  at  the  reduced  pressure.    If  the  temper- 
ature is  not  the  same,  within  o°.5,  the  observed  pressure  should 
be  corrected  as  in  Exp.  8. 

(c)  Fill  the  flask  with  dry  carbon  dioxide  at  atmospheric 
pressure.    This  can  best  be  done  by  pumping  out  the  flask  and 
admitting  the  gas  from  the  generator  several  times  in  succes- 
sion.   Take  care  not  to  allow  any  air  to  pass  through  the  acid 
into  the  generator;  and  keep  the  stop-cock  closed  when  not 
using  the  generator.    When  the  flask  is  filled  with  carbon  diox- 
ide at  a  known  pressure  and  temperature,  weigh  it  as  before. 

(d)  Pump  the  carbon  dioxide  out  until  a  low  pressure  is 
obtained,  as  in  the  case  of  the  air,  and  weigh  again. 

(e)  By  the  use  of  equation  (3),  calculate  from  your  results 
the  relative  density  of  carbon  dioxide  with  respect  to  air,  un- 
der the  given  conditions  of  temperature  and  pressure  prevail- 
ing in  the  room.     How  would  you  proceed  to  apply  Boyle's 
and  Charles'  laws  to  the  result  in  order  to  find  what  the  ratio 
would  be  under  standard  conditions? 

10.     UNIFORMLY   ACCELERATED   MOTION. 

The  purpose  of  this  experiment  is  to  determine  the  acceler- 
ation of  a  freely  falling  body  from  the  trace  made  by  a  vi- 
brating tuning  fork  in  touch  with  the  body  while  falling. 


10]  UNIFORMLY  ACCELERATED   MOTION.  25 

Conceive  of  a  body  moving  in  a  straight  line  with  uniformly 
accelerated  motion,  being  at  a  point  A0  at  a  certain  time,  A^ 
one  interval  of  time  later,  A2  at  the  end  of  the  second  in- 
terval of  time,  etc.  Let  sl  be  the  distance  covered  during  the 
first  interval  of  time,  s2  the  distance  covered  during  the  sec- 
ond interval,  etc.  ;  and  let  t  be  the  number  of  seconds  in  the 
given  interval  of  time.  Let  v±  be  the  average  velocity  of  the 
body  during  the  first  interval,  vz  that  during  the  second  inter- 
val, etc.  Then,  by  the  definition  of  average  velocity,  we  have 

(i)  ^  =  ^'     *>,  =  -'     etc. 

Let  flj  be  the  average    acceleration    between  the  first  and  sec- 

ond intervals  of  time,  a2  that  between  the  second  and  third 

intervals,   etc.     Then,   by  the  definition   of  average  accelera- 
tion, we  have 


or  substituting  from  (i),  we  get 

x     \  «*o  «i  **'-l  •**> 

(3)  tfi  =  JL7r-L>     «,  =  -i-?-^»     etc. 

If  the  body  has  uniformly  accelerated  motion,  alt  a2,  etc.  must 
be  equal. 

(If  the  motion  is  uniformly  accelerated,  the  velocity  will  increase 
at  a  constant  time-rate.  Then  vv  the  average  velocity  for  the 
first  interval  of  time,  will  be  equal  to  the  instantaneous  velocity  of 
the  body  at  the  middle  instant  of  that  interval;  similarly  v2  will 
be  equal  to  the  instantaneous  velocity  at  the  middle  instant  of  the 
second  interval  of  time,  etc.  Between  the  middle  instants  of  any 
two  successive  time-intervals  the  time  elapsing  is  evidently  equal 
to  t.  The  average  acceleration,  then,  between  the  middle  instants 
of  the  first  and  second  intervals  is  (v2  —  v^)/t>  as  given  above.) 

(a)  There  are  two  forms  of  apparatus  used  in  the  labora- 
tory for  this  experiment.  In  one  form,  the  falling  body  con- 


26  CENTRIPETAL    FORCE.  [ll 

sists  of  a  brass  frame  which  falls  about  120  cm.  along  ver- 
tical guides  which  offer  very  little  friction.  This  frame  car- 
ries with  it  a  tuning  fork,  one  prong  of  which  is  provided 
with  a  stylus  which  traces  a  wavy  line  upon  the  whitened  glass 
plate  clamped  vertically  in  the  support.  The  release  of  the 
fork  by  the  lever  at  the  top  causes  the  prongs  to  vibrate. 

In  the  other  form,  the  fork  is  stationary  and  the  glass  plate, 
upon  which  the  trace  is  to  be  made,  falls  about  50  cm.  along 
the  vertical  guides.  The  vibrations  of  the  fork,  in  this  case, 
are  maintained  electrically.  In  both  forms,  a  plumb  line  is 
used  to  adjust  the  plate  and  guides  for  the  fork,  so  that  they 
will  be  accurately  vertical.  The  plate  is  first  covered  with  a 
thin  coat  of  corn-starch  and  alcohol,  which  quickly  dries.  It  5s 
then  placed  in  the  frame  and  adjustments  made.  At  least  three 
good  traces  should  be  obtained.  A  fine  line  is  next  ruled  along 
one  edge  of  the  trace;  and,  starting  at  any  convenient  point, 
points  four  or  five  vibrations  apart  are  marked  off  and  their  dis- 
tances apart,  slf  s2,  etc.,  measured.  Tabulate  these  values  of  s 
and  their  successive  differences.  Repeat  for  points  ten  vibra- 
tions apart.  These  measurements  should  be  made  for  at  least 
two  traces. 

(fr)  From  the  known  value  of  the  frequency  of  the  fork 
find  t.  Calculate  the  acceleration  for  each  set  of  observations 
and  take  the  mean.  Is  it  constant?  Estimate  the  precision 
of  measurement  of  the  result.  Name  the  principal  sources  of 
error. 


11.    CENTRIPETAL  FORCE. 
References. — Duff,  pp.  24,  35;   Millikan,  p.   100. 

The  object  of  this  experiment  is  to  determine  the  force 
necessary  to  keep  a  body  of  given  mass  in  a  circle  of  given 
radius,  while  it  moves  with  constant  speed.  Experience 
shows  that  a  body  in  motion  will  continue  to  move  with  the 


H]  CENTRIPETAL    FORCE.  27 

same  speed  in  the  same  straight  line,  unless  acted  upon  by  some 
outside  force.  An  outside  force,  if  acting  in  the  direction  of 
the  motion,  will  cause  a  change  in  speed;  if  acting  at  right 
angles  to  the  direction  of  motion,  it  will  cause  no  change  in 
speed,  but  will  cause  a  change  in  the  direction  of  the  motion.  A 
body  in  motion  always  moves  in  a  straight  line,  unless  there 
is  a  force  applied  causing  it  to  leave  the  straight  line.  If 
the  force  perpendicular  to  the  line  of  the  motion  be  momentar- 
ily supplied,  the  direction  of  the  motion  is  changed,  but  the 
body  continues  to  move  in  a  straight  line  at  an  angle  with 
its  former  direction.  If  the  force  be  continuously  supplied,  the 
body  moves  in  a  curved  path.  If  the  body  be  kept  in  a  circu- 
lar path,  a  force  of  definite  magnitude  must  be  continuously 
applied  to  the  body,  the  direction  of  the  force  being  always  per- 
pendicular to  the  instantaneous  direction  of  the  motion. 
Since  the  instantaneous  direction  of  motion  is  along  the  tan- 
gent, the  force  perpendicular  to  the  direction  of  motion  must 
be  along  the  radius  of  the  circle.  If  the  force  ceases  to  be 
supplied,  the  body  ceases  to  leave  the  straight  line  and  hence 
continues  to  move  in  the  tangent  to  the  circle  at  the  position 
occupied  by  the  body  at  the  instant  the  force  ceased  to  act. 
This  is  illustrated  by  whirling  a  stone  at  the  end  of  a  string — 
the  string  supplies  the  force  necessary  to  keep  the  stone  in 
a  circular  path.  If  the  string  breaks,  the  necessary  force  is 
no  longer  supplied,  and  the  stone  is  no  longer  pulled  out  of 
the  straight-line  path.  It  moves  away,  therefore,  along  a  tan- 
gent to  its  former  circular  path.  This  central  force  is  called 
the  Centripetal  Force,  or  the  Normal  Force.  It  is  called  the 
normal  force  because  it  is  always  normal  to  the  curved  path. 
It  is  always  directed  toward  the  concave  side  of  the  curve. 
If  the  path  is  a  circle,  it  is  directed  inward  along  the  radius. 
The  acceleration  which  it,  as  an  unbalanced  force,  gives  the 
body  is  also  inward  along  the  radius  and  is  called  Normal  Ac- 
celeration. 


28  CENTRIPETAL    FORCE.  [ll 

For  a  circular  motion  the  magnitude  of  the  normal  acceler- 
ation is  equal  to  v2/r,  where  v  is  the  speed  of  the  body  and  r 
is  the  radius  of  the  circle.  By  the  Force  Equation  an  unbal- 
anced force  acting  upon  a  body  is  proportional  to  the  product 
of  the  mass  m  of  the  body  and  the  acceleration  produced.  We 
have,  then, 

v'i 
Normal  acceleration  (#„)  =  - » 

vi 
Centripetal  force  =  kman  =  km  —  • 

If  the  quantities  m,  v,  and  r  are  expressed  in  C.  G.  S.  units 
the  factor  k  will  be  unity  and  the  force  will  be  given  in  dynes. 

(a)  To  a  rotator  is  attached  the  "centripetal  force"  appar- 
atus. Two  masses,  m±  and  w2,  are  arranged  to  slide  along 
the  horizontal  guides.  They  are  attached,  by  means  of  cords 
passing  over  pulleys,  to  a  large  mass  M,  which  can  slide 
up  and  down  along  the  vertical  rod.  As  the  speed  of  rota- 
tion is  increased,  more  and  more  force  must  be  supplied  to 
ml  and  m2  in  order  to  hold  them  to  a  circular  path.  Finally, 
when  the  speed  passes  a  certain  value,  the  force  necessary  to 
keep  the  masses  moving  in  their  circular  paths  is  greater  than 
the  weight  of  M  can  supply,  so  the  mass  M  is  lifted. 

The  speed  may  be  so  regulated  that  M  remains  about  half- 
way up  the  rod,  or  better,  slowly  rises  and  falls  past  this  point. 
Its  weight,  Mg  dynes,  represents  the  normal  or  centripetal 
force  supplied  to  the  masses  m±  and  m2.  Write  the 
equation  representing  this  relation.  The  masses  M,  mlf 
and  w2,  must  be  determined,  and  the  distances  of  ml 
and  m2  from  the  axis  of  rotation.  The  speeds  of  m±  and  m.2 
may  be  calculated,  provided  the  number  of  rotations  in  a 
given  time  be  counted.  Make  several  trials,  selecting  each 
time  a  different  set  of  values  of  the  masses  or  of  their  dis- 
tances from  the  axis.  In  each  case  maintain  the  speed  for 
five  minutes  or  more. 


12]  THE   PRINCIPLE   OF    MOMENTS.  2Q 

(b)  For  each  trial,  test  the  equality  of  the  weight  Mg  and 
the  calculated  centripetal  force  required,  and  determine  the 
percentage  difference.  Point  out  the  principal  sources  of  error 
in  the  experiment. 

In  the  case  of  a  body  in  circular  motion  what  term  is  com- 
monly applied  to  the  reaction  against  the  centripetal  force? 
Does  it  act  on  the  body,  or  not? 

In  the  case  of  a  skater  describing  a  circle  on  ice,  what  sup- 
plies the  needed  centripetal  force?  Are  the  radial  forces  act- 
ing on  the  skater's  body  balanced?  Are  the  vertical  ones  bal- 
anced ? 

12.     THE  PRINCIPLE  OF  MOMENTS. 
References. — Millikan,   p.   29;   Duff,   pp.   78-81. 

The  purpose  -of  this  experiment  is  to  determine  the  condi- 
tion which  must  be  satisfied  if  a  body,  acted  upon  by  three 
or  more  forces  in  the  same  plane,  is  to  remain  in  equilibrium 
with  reference  to  rotation.  In  order  that  a  body  at  rest  shall 
remain  at  rest,  or  a  body  in  motion  remain  in  motion  with  con- 
stant linear  and  angular  velocity,  the  vector  sum  or  resultant  of 
all  the  forces  acting  upon  it  must  be  zero,  and  the  algebraic  sum 
of  the  moments  of  these  forces  about  any  axis  must  be  zero. 
In  the  case  where  all  the  forces  are  in  the  same  plane,  the  sec- 
ond of  these  conditions,  sometimes  called  the  Principle  of  Mo- 
ments, requires  that  the  sum  of  the  moments  of  all  the  forces 
about  any  and  every  point  selected  in  the  plane  as  a  center  of 
moments  shall  be  zero.  For  instance,  let  us  suppose  a  <case 
where  three  forces,  Flt  F2,  F3,  act  in  the  same  plane  upon  a 
given  body  and  the  body  remains  in  equilibrium  with  respect 
to  rotation ;  if  C  be  any  point  in  the  plane  and  llt  12  and  /3  be 
the  lever-arms  or  perpendicular  distances  from  C  to  the  lines 
of  direction  of  the  three  forces,  and  if  the  moments  with  re- 
spect to  C  of  two  of  the  forces  be  anti-clockwise  or  positive  and 


30  THE   PRINCIPLE  OF    MOMENTS.  [l2 

the  moment  of  the  third  force  with  respect  to  C  be  clockwise 
or  negative,  then  the  Principle  of  Moments  requires  that 


To  prove  the  principle  it  is  only  necessary  to  show  that  the 
sum  is  zero  for  one  selected  point,  provided  that  this  point  is 
so  chosen  as  not  to  lie  in  the  line  of  any  of  the  forces.  If  a 
point  in  the  line  of  any  force  were  chosen,  the  moment  of  that 
force  with  reference  to  that  point  as  a  center  of  moments 
would  be  zero  no  matter  what  the  value  of  the  force  ;  hence, 
the  result  for  such  a  choice,  would  not  be  a  test  of  the  prin- 
ciple. 

The  apparatus  used  to  test  the  principle  consists  of  a  circu- 
lar table  with  a  movable  disk  resting  on  bicycle  balls.  The 
disk  may  be  pivoted  in  the  center  if  desired.  To  pegs,  placed 
at  will  in  the  disk,  cords  are  attached  which  pass  over  pulleys 
clamped  at  different  points  around  the  circular  table.  From 
the  ends  of  the  cords  are  suspended  known  masses  whose 
weight  produces  the  forces  required. 

(a)  Place  the  disk  on  four  bicycle  balls,  widely  separated, 
and  level  up  the  table  so  that  the  disk  will  not  tend  to  move  in 
any  one  direction  in  preference  to  another.     Pivot  the  disk  in 
the  center  and  place  a  sheet  of  manila  paper  upon  it.     Attach 
cords  to  it  at  three  different  points  chosen  at  random;  and, 
placing  the  pulleys  at  any  convenient  points,  add  masses  until 
the  three  forces  are  of  convenient  values.     See  that  the  disk  is 
free  to  move  on  the  bicycle  balls  and  that  the  cords  all  lie  in  a 
plane  close  to  and  parallel  to  the  top  of  the  disk;  then  mark 
points  or  lines  on  the  paper  to  indicate  the  directions  of  the 
forces.     Note  the  magnitude  of  the  forces,  counting  in  the 
weight  of  the  pan  in  each  force. 

(b)  Remove  the  paper,  trace  the  lines  of  direction  of  the 
forces,  and  make  the  measurements  necessary  to  determine 
-their  moments  about  the  pivot  as  an  axis.     Find  the  sum  of 


13]  THE   SIMPLE    PENDULUM.  $1 

the  moments,  taking  those  as  positive  which  tend  to  produce 
a  counter-clockwise  rotation  about  the  given  axis,  and  those 
as  negative  which  tend  to  produce  a  clockwise  rotation. 

(c)  Choose  at  random  any  point  on  the  paper  used  in  (a) 
and   (b),  and  find  the  sum  of  the  moments  about  this  point 
as  a  center  of  moments.    Why  is  not  the  sum  zero? 

(d)  Remove  the  pivot,  and  repeat  (a)  and  (b)  once,  se- 
lecting in  turn  as  centers  of  moments  three  points  as  widely 
separated  as  possible.     Keep  the  disk  free  of  the  rim  about  it, 
so  that  the  only  forces  acting  on  the  disk  in     the    horizontal 
plane  are  those  due  to  the  cords.     Find  the  sum  of  the  mo-  V' 
ments  for  each  of  these  centers  as  before.     Also  find  the  vec- 
tor sum  or  resultant  of  the  forces  by  the  method  of  the  closed 
polygon. 

(e)  Repeat  (d),  using  four  forces  instead  of  three. 

(/)  From  the  data  of  (a)  and  (b)  determine  the  vector  sum 
of  the  three  forces  used  in  that  case.  If  this  sum  is  not  zero, 
it  means  that  the  pivot  itself  exerted  a  force  on  the  disk  in 
the  same  plane  with  the  three  forces.  What  do  you  conclude 
is  the  magnitude  and  direction  of  this  force?  Draw  its  line 
of  direction  on  the  paper,  and  then  repeat  (c},  including  now 
in  your  sum  the  moment  of  the  force  due  to  the  pivot.  Is 
the  sum  now  approximately  zero? 

(g)  In  the  various  cases  of  equilibrium  considered  above, 
what  do  you  find  the  vector  sum  of  the  forces  to  be  ?^  What 
have  you  found  to  hold  true  for  the  moments  of  these  forces  ? 
Calculate  the  percentage  error  for  one  case. 


13.     THE  SIMPLE  PENDULUM. 
References. — Duff,  p.  87;  Millikan,  p.  95. 

The  purpose  of  this  experiment  is  to  determine  the  acceler- 
ation due  to  gravity  from  a  knowledge  of  the  period  and  length 


32  THE    SIMPLE    PENDULUM  [l3 

of  a  simple  pendulum.     For  vibrations  of     small     amplitude 
the  period  of  such  a  pendulum  is  given  by  the  equation, 


where  T  is  the  time  of  one  complete  vibration,  /  is  the  length 

of  the  pendulum,  and  g  is  the  acceleration  due  to  gravity.     Tf 

T  and  /  are  known  for  any  place,  g  can  be  determined  for  that 

place. 

Method  of  Coincidences. 

In  the  present  experiment,  T  is  to  be  measured  by  compar- 
ing, by  the  ''method  of  coincidences,"  the  period  of  the  simple 
pendulum  with  that  of  a  clock  pendulum  of  known  period.  An 
electric  circuit  is  completed  through  an  electric  bell,  the  clock 
pendulum,  the  simple  pendulum,  and  the  mercury  contacts  at 
the  bottom  of  each  pendulum.  Assume  that  the  period  of  the 
clock  pendulum  is  two  seconds,  that  is,  that  the  time  of  a  single 
swing  or  half-vibration  is  one  second.  If  the  period  of  the 
simple  pendulum  were  the  same  and  the  two  pendulums  be 
started  together,  they  would  vibrate  in  coincidence  and  the 
bell  would  ring  with  every  passage.  If,  however,  the  time  of 
a  single  swing  of  the  simple  pendulum  were  less  than  one 
second,  say  by  i/w  th  of  a  second,  it  would  gain  on  the  clock 
pendulum  and  thus  fall  out  of  coincidence  with  it,  so  that 
the  bell  would  cease  to  ring  until  n  seconds  later,  when  the 
two  pendulums  would  be  in  coincidence  again.  Let  us  sup- 
pose that  the  time  between  these  successive  coincidences  is 
100  seconds,  then  we  know  that  in  this  time  the  clock  pen- 
dulum has  made  one  hundred  half-  vibrations  and  the  simple 
pendulum  one  more,  or  101  half-vibrations.  In  other  words, 
the  simple  pendulum  has  made  101  half-vibrations  in  100  sec- 
onds, hence  the  value  of  its  half-period  is  100/101  seconds. 
If,  on  the  other  hand,  the  simple  pendulum  had  been  observed 
to  lag  behind  the  clock  pendulum,  and  the  time  between  sue- 


13]  THE  SIMPLE  PENDULUM.  33 

cessive  coincidences  remained  the  same,  we  would  know  that 
its  half-period  is  100/99  seconds. 

(a)  The  simple  pendulum  used  consists  of  a  brass  sphere 
suspended  from  a  knife-edge  by  a  wire  so  that'  the  length  is 
adjustable.     The  mercury  contact  below  should  be  so  adjusted 
that  the  platinum  point  on  the  ball  passes  freely  through  it 
Adjust  the  pendulum  so  that  its  length  is  either  greater  or  less, 
by  2  or  3  cm.,  than  that  of  a  pendulum  beating  seconds.    Two 
different    lengths    (in    successive   determinations)    should   be 
used  such  that  one  is  greater  and  the  other  less  than  that  of 
a  pendulum  beating  seconds.     In  getting  the  length  it  is  well 
to  measure  with  a  meter  rod  and  square  to  the  top  of  the  ball, 
and  then  to  determine  the  diameter  of  the  ball  with  the  cali- 
pers.    The  length  of  the  pendulum  is  the  distance  from  the 
knife-edge  to  the  center  of  the  ball.     After  adjusting  the  mer- 
cury contact,  start  the  ball  swinging  in  an  arc  of  about   10 
cm.,  taking  care  not  to  give  it  a  twisting  vibratory  motion. 
During  the  vibrations  watch  the  hands  of  the  laboratory  clock 
and  record  the  hour,  minute  and  second  of  each   successive 
coincidence  between  the  simple  pendulum  and  the  clock  pen- 
dulum, up  to  ten  or  more.     If  the  bell  rings  for  more  than  one 
swing  during  each  coincidence,  take  the  mean  of  the  times  of 
the  first  and  last  rings  as  the  time  of  the  coincidence. 

(b)  To  obtain  from  the  data  a  more  reliable  average  value 
of  the  time  between  successive  coincidences,  proceed  as  fol- 
lows :     Find  the  difference  in  time  between  the  first  and  sixth 
coincidences,  the  second  and  seventh,  and  so  on,  and  take  the 
mean.     From  this  the  average  time  between  successive  coin- 
cidences may  be  found  and  the  period  calculated.     Be  sure  to 
note  whether  the  pendulum  was  gaining  or  losing  on  the  clock. 
Calculate  the  value  of  g  for  the  two  cases  and  take  the  mean. 

(c)  What  effect  would  be  produced  upon  the  vibration  of 
a  pendulum  by  carrying  it,  (i)  to  a  mountain  top,  (2)  from 
the  equator  to  the  pole  of  the  earth?    In  what  way  does  the 


34  THE  FORCE  EQUATION.  [14 

pendulum  used  in  this  experiment  fall  short  of  the  require- 
ments for  a  simple  pendulum?  What  is  the  object  of  taking 
a  small  amplitude  of  vibration? 


14.    THE  FORCE  EQUATION. 
References. — Duff,  p.  31;   Millikan,  p.    15. 

If  F  is  the  resultant  force  acting  on  a  body,  m  its  mass,  and 
a  the  acceleration  produced,  we  have,  as  a  result  of  definition 
and  experiment, 

(i)     F  =  k  m  a. 

This  equation  is  called  the  Force  Equation,  or  Equation  of 
Motion,  and  the  purpose  of  the  experiment  is  to  verify  it. 
The  factor  k  in  the  equation  is  a  numerical  constant  whose 
value  depends  upon  the  system  of  units  used.  This  equation 
states  ( i )  that,  if  two  forces  act  on  bodies  of  the  same  mass, 
the  accelerations  produced  will  be  directly  proportional  to  the 
forces;  and  (2)  that,  if  two  forces  produce  the  same  acceler- 
ation in  two  bodies  of  different  mass,  the  masses  will  be  di- 
rectly proportional  to  the  forces.  Let  M,  M  be  two  equal 
masses  suspended  from  a  cord  passing  over  a  pulley  whose 
friction  and  rotational  inertia  we  will  assume  to  be  negligi- 
ble. The  total  mass  suspended  is  2.M ;  the  resultant  force  act- 
ing upon  it  is  zero.  Let  a  mass  m  be  removed  from  one  side. 
The  resultant  force  F^  now  is  k  m1  g,  and  it  will  cause  the 
mass  (zM  —  m±)  to  move  in  its  direction  with  an  acceleration 
Oj ;  hence,  by  equation  ( i ) , 

(2)      F,  =  k  (2M  —  m,)  a,. 

If  a  different  force  be  applied  by  changing  to  w2  the  value  of 
the  mass  removed,  the  resultant  force  (F2)  will  be  k  m2  g, 


14]  THE  FORCE  EQUATION.  35 

and  it  will  produce  an   acceleration   a^  ;  hence,  by  equation 


Hence 


(3)      F2  =  fe  (2M  —  w2) 
~  **•>*' 


- 

I       k  (2M  — 

2^ 


An  experimental  verification  of  equation  (4)  will  constitute 
a  verification  of  equation  (i),  though  it  will  not,  of  course, 
determine  the  value  of  the  constant  k. 

The  apparatus  used  in  Exp.  10  is  employed,  with  the  addi- 
tion of  a  pulley-attachment  at  the  top  over  which  a  cord  passes, 
from  one  end  of  which  the  fork  or  the  glass  plate  (dependent 
upon  which  form  of  apparatus  is  used)  is  suspended  and  from 
the  other  end  a  number  of  masses  just  sufficient  to  balance  the 
same  and  the  friction  of  the  pulley.  Note  the  precautions  given 
in  Exp.  10.  Special  care  should  be  taken  to  insure  as  little 
friction  as  possible. 

(a)  Adjust  the  apparatus  so  that  a  good  trace  may  be  ob- 
tained and  so  that  a  slight  tap  will  cause  the  fork  or  the  glass 
plate  to  descend  without  acceleration.     The  forces,  including 
friction,  are  then  just  balanced.     Cover  the  plate  with  a  thin 
coat  of  corn-starch  and  alcohol.     Take  care  to  have  the  sty- 
lus exert  the  same  pressure  against  the  plate  throughout  the 
experiment. 

(b)  Remove  a  mass  m1  from  the  balancing  masses.    Note 
the  total  mass   (2M  —  mt)   of  the  moving  system.     Obtain 
two  good  traces. 

(c)  Repeat  with  a  different  mass  m2  removed,  the  total 
mass  of  the  system  now  being  (2M  —  w2). 

(d)  Repeat  again  with  a  third  mass  removed. 

(e)  Measure  the  traces  as  explained  in  Exp.   10,     using 
five  vibrations  of  the  fork  as  the  interval  of  time.     Calculate 


36  SURFACE  TENSION  BY  JOLLY^S  BALANCE.  [l$ 

the  accelerations  a^  a2,  a3,  corresponding  to  (b),  (c),  (d) 
above.  Then  make  two  tests  of  equation  (4)  by  substituting  in 
the  same.  Calculate  the  percentage  difference  between  the  two 
sides  of  the  equation  in  each  case. 

If  the  masses  removed  in  (&),  (c),  (d)  had  simply  been 
transferred  from  one  side  of  the  pulley  to  the  other,  what 
changes  would  be  required  in  substituting  in  equation  (4)  ? 


15.  SURFACE  TENSION  BY  JOLLY'S  BALANCE. 

References.— Duff,  p.  146;  Millikan,  p.   181. 

The  purpose  of  this  experiment  is  to  obtain  a  direct  measure 
of  the  surface  tension  of  a  liquid  by  balancing  it  against  the 
tension  in  a  stretched  spring.  A  wire  rectangle  is  hung  from 
the  spring  of  a  Jolly's  balance  and  allowed  to  dip  in  a  soap  solu- 
tion which  forms  a  film  across  the  rectangle.  When  equili- 
brium is  established  the  force  due  to  surface  tension  in  the 
two  surfaces  of  the  film  must  just  balance  the  tension  in  the 
spring.  By  knowing  the  force  which  will  stretch  the  spring 
the  same  amount,  we  have  a  measure  of  the  total  force  due  to 
surface  tension.  If  T  is  the  value  of  the  surface  tension  per 
centimeter  width  of  the  film,  /  the  width  of  the  rectangle  along 
the  surface  of  the  liquid,  and  F  the  force  exerted  by  the  spring, 
then  it  follows,  because  the  forces  are  in  equilibrium,  that 

F  =  2/T 

Knowing  F  and  /,  the  value  of  T  can  thus  be  found. 

The  Jolly's  balance  used  is  one  of  the  two  forms  used  in 
Exp.  2.  Ask  for  directions,  if  its  operation  is  not  already  un- 
derstood. Wire  rectangles  of  different  sizes  and  a  wide  beaker 
are  provided.  The  greatest  care  must  be  taken  that  the 
beaker  and  rectangles  are  clean.  They  should  be  washed  in 
caustic  potash  and  rinsed  thoroughly  in  hot  water  before  be- 
ing used  and  before  changing  to  another  liquid.  Do  not  touch 


l6]  CAPILLARITY.      RISK  OT?  LIQUIDS    IN    TUBES.  37 

with  the  fingers  the  inside  of  the  beaker,  the  liquid,  or  the  part 
of  the  rectangle  on  which  the  film  is  formed. 

(a)  Suspend  a  rectangle,  2  cm.  wide,  from  the  spring,  and 
let  it  be  partially  immersed  in  a  beaker  of  soap-solution.  Read 
the  extension  of  the  spring  when  there  is  no  film  in  the  rect- 
angle, and  again  with  a  film  across  it.     The  rectangle  should 
be  immersed  to  the  same  depth  in  the  two  cases,  so  as  to  elim- 
inate the  effect  of  the  buoyancy  of  the  liquid.     Take  three  sets 
of  readings.     Note  whether  the  pull  of  the  film  depends  upon 
the  area  of  it  formed  in  the  rectangle. 

Repeat  these  measurements,  using  rectangles  4  cm.  and  6 
cm.  wide. 

(b)  Calibrate  the  balance  by  observing  the  extension  pro- 
duced by  known  standard  masses. 

(c)  Use  the  rectangle,  4  cm.  wide,  cleaning  it     and    the 
beaker  thoroughly,  and  repeat  (a)  with  water  fresh  from  the 
tap.     As  a  film  of  no  appreciable  height  will  form  with  pure 
water,  take  the  reading  of  the  balance  without  the  film  when 
the  under  side  of  the  upper  wire  of  the  rectangle  is  just  above 
the  surface  of  the  water  and  not  in  contact  with  it ;  and  again, 
after  immersing  the  upper  wire  of  the  rectangle  so  as  to  wet 
it,  take  a  reading  when  it  breaks  away  from  the  surface.  Take 
three  sets  of  readings. 

(d)  Repeat  (c),  using  water  at  5O°C.  or  higher. 

(e)  Repeat  (c),  using  alcohol. 

(/)  From  the  data  taken  in  (a),  state  how  the  total  tension 
in  the  film  varies  with  its  width.  Calculate  the  surface  ten- 
sion, T,  in  dynes  per  cm.,  for  the  liquids  used  in  (a),  (c), 
(d),  and  (e),  comparing  the  values  obtained  and  pointing  out 
how  the  surface  tension  is  affected  by  the  temperature. 

16.     CAPILLARITY.     RISE  OF  LIQUIDS  IN  TUBES. 

Reference. — Duff,    p.    149. 

In  the  present  experiment  the  values  of  the  surface  tension 
of  water  and  of  alcohol  are  to  be  measured  by  observing  the 


38  CAPILLARITY.      RISE  OF  LIQUIDS  IN  TUBES.  [l6 

rise  of  these  liquids  in  capillary  tubes.  When  the  inner  sur- 
face of  a  tube  is  wet  by  a  liquid,  the  surface  tension  of  the 
latter  may  be  considered  as  acting  upward  at  all  points  around 
the  circumference  of  the  tube.  The  total  vertical  component 
of  this  force  is  2-n  r  T  cos  a,  where  r  is  the  radius  of  the  tube, 
T  the  surface  tension  in  dynes  per  cm.,  and  a  is  the  angle  of 
contact  between  the  liquid  and  the  tube.  If  the  tube  is  of 
small  bore,  the  liquid  will  rise  inside  the  tube,  equilibrium  be- 
ing established  when  the  weight  of  the  liquid  within  the  tube 
above  the  level  of  the  liquid  outside  equals  the  vertical  force 
upward  due  to  surface  tension.  If  d  is  the  density  of  the  li- 
quid, h  its  height  in  the  capillary  tube  above  the  surface  level, 
and  g  the  acceleration  due  to  gravity,  it  follows  that 

7rr2hdg  =  27rrT  COS  a. 

From  this  equation  the  value  of  T,  the  surface  tension  in  dynes 
per  cm.,  can  be  found. 

(a)  Capillary  tubes  of  different  sizes  are  provided.  These 
may  be  thermometer-tubes  or  larger  glass  tubing  drawn  out 
to  a  fine  bore.  In  either  case  every  precaution  must  be  taken 
to  have  the  tubes  perfectly  clean  and  free  from  all  traces  of 
grease.  They  should  be  cleaned  with  caustic  potash  solution, 
rinsed  with  tap  water  and  then  with  the  liquid  to  be  experi- 
mented with  (in  this  case,  water).  'With  a  rubber  band  fasten 
the  tubes  side  by  side  to  a  glass  scale,  and  place  the  scale  and 
tubes  vertically  in  a  small  dish  of  distilled  water.  Lower  the 
tubes  first  to  the  bottom  of  the  dish  so  as  to  wet  the  inside  for 
some  distance  above  the  point  to  which  the  water  will  rise. 
Then  clamp  them  with  the  ends  below  the  surface,  and  note  on 
the  scale  the  point  to  which  the  water  rises  in  each  tube.  To 
obtain  the  reading  for  the  water-surface  in  the  dish  a  wire 
hook  is  provided,  which  should  be  brought  up  so  that  the  point 
is  just  even  with  the  surface.  Then  read  the  height  of  this 
point  on  the  glass  scale. 


17]  RISE   OF    LIQUIDS   BETWEEN    PLATES.  39 

(b)  Measure  the  inside  diameter  of  the  tube  with  a  microm- 
eter microscope.     If  drawn-out  tubing  is  used,  scratch  the 
tube  with  a  file  at  the  point  to  which  the  water  rises,  break 
it  and  measure  the  diameter  of  the  end.  If  the  tube  is  uniform 
in  bore,  its  diameter  can  be  found  either  with  the  micrometer 
microscope,  or  by  means  of  a  thread  of  mercury  drawn  into 
the  tube.     In  case  the  latter  method  is  used,  the  length  and 
mass  of  the  thread  and  the  density  of  mercury  are  all  the  data 
needed  for  calculating  the  diameter.     Calculate     the     surface 
tension  of  water,  and  compare  this  value  with  that  found  in 
Kxp.  15.    For  pure  water  and  ordinary  glass  the  angle  of  con- 
tact is  approximately  zero. 

(c)  In  the  same  way  find  the  surface  tension  of  alcohol. 
For  the  angle  of  contact  in  this  case  see  the  Tables. 

Would  the  water  or  alcohol  rise  as  high  in  the  tubes  if  the 
experiment  were  performed  in  a  vacuum?  Explain. 

If  a  thread  of  water  were  placed  in  a  horizontal,  conical- 
shaped  tube,  in  which  direction  along  the  tube  would  it  move? 
Explain.  If  mercury  instead  of  water  were  used,  what  would 
happen,  and  why? 

17.     RISE   OF  LIQUIDS   BETWEEN  PLATES. 
Reference. — Hastings   and    Beach,   p.    146. 

In  the  present  experiment  the  surface  tension  of  water  and 
of  alcohol  is  to  be  measured  by  means  of  the  rise  of  the  liquid 
in  a  wedge-shaped  space  between  two  plates  of  glass.  The  two 
plates  of  glass,  which  are  in  touch  with  each  other  on  one  side, 
are  separated  on  the  other  side  by  a  thin  piece  of  brass  placed 
between  the  opposite  edges  of  the  plates.  The  plates  are 
clamped  together  and  placed  upright  in  a  shallow  dish  of  li- 
quid. If  the  liquid  wets  the  plates,  it  will  rise  in  the  wedge- 
shaped  space,  forming  a  smooth  curve  which  extends  from 
the  surface  of  the  liquid  in  the  dish,  on  the  side  where  the 


4O  RISE   OF   LIQUIDS    BETWEEN    PLATES.  [17 

plates  are  farthest  apart,  to  a  point  high  above  this  level,  on 
the  other  side  where  the  plates  are  in  touch  with  each  other. 
The  general  effect  is  similar  to  that  obtained  by  a  row  of  small 
tubes  of  gradually  decreasing  bore.  We  may  consider  that  at 
some  point  along  the  curve  a  thin  vertical  slice  or  rectangular 
prism  of  the  liquid  is  taken.  Let  d,  the  distance  between  the 
plates  at  the  point  chosen,  be  the  width  of  the  prism;  .r  (very 
small),  the  thickness  of  the  prism  in  a  direction  parallel  to  the 
plates  and  to  the  surface  of  the  liquid  in  the  dish ;  and  //  the 
height  of  the  prism  above  the  surface  of  the  liquid  in  the  dish. 
The  surface  tension  which  acts  upon  this  prism  evidently  has 
a  vertical  component  upward  equal  to  2  T  x,  where  T  is  the 
value  of  the  surface  tension  in  dynes  per  cm.  This  force  must 
equal  the  weight  of  the  prism  of  liquid  which  is  h  x  d  D  g, 
where  D  is  the  density  of  the  liquid  and  g  the  acceleration  due 
to  gravity.  From  this  relation  T  can  be  found. 

(a)  Clean  the  plates  very  carefully  with  caustic  potash 
solution,  and  rinse  with  water.  Clamp  them  together  as  indi- 
cated above,  and  upon  one  side  of  one  of  the  plates  place  a  thin 
sheet  of  white  paper.  Stand  the  plates  upright  in  a  shallow 
vessel  of  distilled  water,  and  looking  through  the  paper  and 
the  plates  toward  the  light,  trace  on  the  paper  the  surface  of 
the  water  between  the  plates,  the  surface  of  the  water  in  the 
dish,  the  outline  of  the  piece  of  metal,  and  the  edge  where  the 
plates  touch  each  other.  (If  the  water-curve  between  the 
plates  is  not  a  smooth  one,  it  will  be  necessary  to  raise  and 
lower  the  plates  in  the  dish  until  the  surfaces  of  the  glass  be- 
tween them  is  thoroughly  wet.)  Removing  the  sheet  of  pa- 
per, draw  a  line  on  the  paper  to  show  the  position  of  the  inner 
edge  of  the  piece  of  brass.  This  line,  as  well  as  the  line  show- 
ing the  position  of  the  edge  where  the  plates  were  in  touch  with 
each  other,  should  be  perpendicular  to  the  line  representing  the 
surface  of  the  water  in  the  dish.  -Select  any  point  P  on  the 
curve  representing  the  surface  of  the  water  between 


l8]  VISCOSITY.       FLOW  OF  LIQUIDS  IN  TUBES.  41 

the  plates.  From  this  point  draw  a  line  perpendicu- 
lar to  the  line  representing  the  surface  of  the  water  in 
the  dish,  and  call  its  length  h.  Let  x  be  an  infinitesimal  dis- 
tance through  P  at  right  angles  to  this  last  line.  To  determine 
the  width  d  between  the  plates  at  P,  proceed  as  follows: 
Draw  a  line  through  P  parallel  to  the  line  representing  the 
surface  of  the  water  in  the  dish  and  let  the  length  along  this 
line  from  P  to  the  line  showing  where  the  plates  were  in  touch 
with  each  other  be  /.  Let  the  whole  distance  from  the  inner 
edge  of  the  piece  of  brass  to  this  same  line  be  L.  Measure  the 
thickness  dl  of  the  piece  of  brass  with  a  micrometer  caliper. 
Then,  at  the  point  P, 


Derive  this  equation.  From  the  values  of  d  and  h  thus  found, 
calculate  the  surface  tension  of  water  in  dynes  per  cm.  Repeat 
the  measurements  and  calculation  for  one  or  two  other  points 
on  the  curve. 

(fr)     Repeat   (a),  using  alcohol  instead  of  water,  and  find 
the  surface  tension  of  alcohol. 


18.    VISCOSITY.    FLOW  OF  LIQUIDS  IN  TUBES. 

Reference. — Duff,    p.    137. 

The  dependence,  of  the  rate  of  flow  in  tubes,  on  the  diameter 
and  length  of  the  tube,  and  on  the  temperature  of  the  liquid 
and  the  kind  of  liquid  used,  is  to  be  observed.  When  a 
liquid  flows  through  a  tube,  if  the  liquid  wets  the  walls  of 
the  tube,  the  layer  of  liquid  in  immediate  contact  with  the  wall 
generally  remains  at  rest.  The  speed  with  which  the  liquid 
moves  increases  from  the  surface  of  the  tube  to  the  axis  of  the 
tube.  Hence,  if  we  imagine  the  liquid  to  consist  of  a  number 
of  hollow  cylinders  coaxial  with  the  tube,  the  fluid  within 


42  VISCOSITY.       FLOW  OF  LIQUIDS  IN  TUBES.  [l8 

each  of  these  cylindrical  shells  will  be  moving  more  slowly 
than  in  the  shell  immediately  inside,  and  more  rapidly  than  in 
the  shell  immediately  outside.  This  relative  motion  of  adjacent 
layers  of  the  liquid  is  determined  by  the  internal  friction  or 
viscosity  of  the  liquid.  Viscosity  varies  greatly  with  the  kind 
of  liquid  used,  this  dependence  upon  the  character  of  the  li- 
quid being  indicated  by  the  coefficient  of  viscosity.  If  a  liquid 
is  very  viscous,  like  syrup,  its  coefficient  of  viscosity  is  high  ; 
if  like  alcohol,  its  coefficient  of  viscosity  is  low.  For  a  given 
liquid  at  a  given  temperature,  the  coefficient  of  viscosity  is  a 
constant. 

In  the  case  of  a  liquid  flowing  through  a  long,  narrow  tube, 
the  volume  V,  issuing  per  second  from  the  end,  depends  upon 
the  difference  in  pressure  p,  between  the  two  ends  of  .the  tube, 
the  radius  r  of  the  tube,  its  length  /,  and  the  coefficient  of  vis- 
cosity c  of  the  liquid.  These  quantities  are  connected  by  the 
relation 


To  compare  the  coefficients  of  viscosity  of  two  different  li- 
quids, it  is  evident,  if  the  above  relation  be  accepted,  that,  for 
the  same  tube  and  equal  times  of  flowing,  the  coefficients  will 
be  in  inverse  proportion  to  the  volumes,  or  cl  :  c2  ==  V2  :  Vy. 

Three  small-bore  tubes  are  provided,  two  being  of  the  same 
length  but  of  different  bores,  and  the  third  being  longer  but 
of  the  same  bore  as  one  of  the  two  shorter  ones.  The  reservojr 
used  consists  of  a  large  bottle  through  whose  cork  are  fitted 
two  glass  tubes,  long  enough  to  reach  about  two-thirds  of  the 
way  to  the  bottom.  The  outside  end  of  one  of  these  tubes 
is  connected  by  rubber  tubing  with  the  tube  through  which 
the  flow  is  to  be  measured  ;  the  other  tube  is  left  open  to  the 
air.  Both  tubes  must  extend  some  distance  below  the  level 
of  the  liquid  in  the  bottle,  and  the  cork  must  be  air-tight.  By 


l8]  VISCOSITY.       FLOW  OF  LIQUIDS  IN   TUBES.  43 

means  of  this  arrangement  a  constant  head  of  pressure  may 
be  obtained.  The  tube,  which  carries  the  liquid  from  the  res- 
ervoir to  the  small-bore  tube,  is  quite  large,  so  that  the  fric- 
tional  resistance  which  it  offers  to  the  flow  will  be  negligible 
as  compared  to  that  offered  by  the  small  tube.  This  makes  it 
reasonable  to  assume  (as  is  done  in  the  experiment)  that  the 
head  of  pressure  is  all  employed  against  the  frictional  resis- 
tance offered  by  the  small  tube. 

(a)  Clean  the  tubes  thoroughly  with  chromic  acid,  and  rinse 
by  drawing  clean  water  through  them  with  a  jet-pump.  Attach 
one  of  the  tubes  to  the  siphon-tube  from  the  reservoir,  letting 
the  lower  end  dip  into  water  in  a  beaker.  Weigh  the  beaker 
and  contained  water  on  the  trip-scales.  Before  replacing  the 
beaker  in  position,  nearly  fill  the  reservoir  with  water  at  the 
room  temperature,  start  the  siphon,  and  let  the  water  run  into 
a  waste  vessel  until  the  air  begins  to  bubble  from  the  lower  end 
of  the  open  tube  up  through  the  water  in  the  reservoir.  Then 
replace  the  beaker,  record  the  height  of  the  water-level  in  it, 
and  allow  the  water  to  flow  for  two  minutes.  Weigh  the  beaker 
again  to  determine  the  volume  which  has  run  through.  The 
head  of  pressure  will  be  given  by  the  difference  in  height  of 
the  lower  end  of  the  open  tube  in  the  reservoir  and  the  mean 
of  the  initial  and  final  levels  in  the  beaker.  Point  out  clearly 
why  the  head  is  measured  from  the  end  of  the  open  tube  and 
not  from  the  water-level  in  the  reservoir.  Make  two  indepen- 
dent trials. 

(&)  Repeat  with  each  of  the  other  tubes.  Measure  the 
diameters  of  the  tubes  with  the  micrometer  microscope,  or  by 
weighing  mercury  which  occupies  a  known  length  of  the  tube. 
What  do  your  results  show  concerning  the  dependence  of  the 
rate  of  flow  on  the  radius  and  length  of  the  tube  ?  Employing 
the  C.  G.  S.  system  of  units,  calculate  the  coefficient  of  vis- 
cosity of  the  water  for  the  three  cases,  and  take  the  average 
value. 


/| /|  EFFLUX  OF  GASES.      RELATIVE  DENSITIES.  [19 

(c)  With  one  of  the  tubes,  use  water  at  5o°-6o°C.  in  the 
reservoir,  and  compare  with  previous  results  to  determine  the 
effect  of  temperature  on  viscosity. 

(d)  Repeat  (c)  with  a  ten-per-cent  solution  of  sugar,  and, 
if  there  is  time,  with  a  ten-per-cent  salt-solution.     Discuss  the 
results,  comparing  them  with  those  of   (a)   and   (&),  noting 
the  effect  upon  viscosity  of  different  sorts  of  dissolved  sub- 
stances. 

19.    EFFLUX  OF  GASES.    RELATIVE  DENSITIES. 
Reference. — Duff,    p.    166. 

The  object  of  this  experiment  is  to  find  the  relative  densities 
of  certain  gases  from  the  observation  of  the  relative  times  of 
efflux  of  equal  volumes  of  these  gases  through  a  small  aper- 
ture. The  ratio  of  the  densities  of  two  gases,  under  the  same 
conditions  as  to  pressure,  is  equal,  very  approximately,  to  the 
inverse  ratio  of  the  squares  of  the  speeds  with  which  the 
gases  escape  through  a  fine  opening  in  a  diaphragm.  Since 
the  time  of  escape  of  a  given  volume  will  be  inversely  as  the 
speed  of  efflux,  it  follows  that  the  ratio  of  the  densities  of  two 
gases  is  equal  to  the  direct  ratio  of  the  squares  of  the  time  of 
efflux  of  equal  volumes  under  the  same  conditions.  This  rela- 
tion was  experimentally  discovered  by  Bunsen.  For  a  proof 
of  it,  from  the  energy  relations,  see  the  reference  given 
above. 

(a)  The  gas-holder  consists  of  a  glass  cylinder,  at  the  top 
of  which  is  a  three-way  stop-cock  and  a  diaphragm  with  a 
fine  opening.  The  cylinder  is  placed  in  a  reservoir  of  mer- 
cury. The  three-way  cock  allows  communication  to  be  made 
with  the  outside  for  filling  or  with  the  diaphragm.  Within  the 
cylinder  is  a  float  which  indicates  when  the  desired  volume  of 
gas  has  escaped. 

First  fill  the  cylinder  with  dry  air.     To  do  this,  turn  the 


2O]  ABSOLUTE  CALIBRATION   OF  A   THERMOMETER.  45 

stop-cock  so  as  to  put  the  cylinder  in  communication  with  the 
air,  and  lower  the  cylinder  as  far  as  it  will  go.  This  drives 
out  most  of  the  contained  gas.  Connect  the  cylinder  with  a 
calcium-chloride  drying-tube,  and  raise  the  cylinder.  This 
operation  will  fill  the  cylinder,  and  by  repeatedly  emptying 
and  filling  the  cylinder  it  will  become  practically  freed  of  the 
moist  air  or  other  gas  previously  contained  in  it.  Close  the 
stop-cock,  and  lowering  the  cylinder,  clamp  it  in  position 
Turning  the  stop-cock  so  that  the  gas  in  the  cylinder  is  in  com- 
munication with  the  diaphragm,  note  the  time  when  the  upper 
point  of  the  float  is  on  a  level  with  the  surface  of  the  mercury 
or  with  a  mark  on  the  cylinder.  Again  note  the  time  when  the 
second  mark  on  the  float  is  on  the  same  level.  Repeat,  mak- 
ing two  or  three  determinations  of  the  time  of  efflux  for  the 
given  volume  of  air,  and  take  the  mean. 

(b)  Repeat  (a),  filling  the  cylinder  with  illuminating  gas, 
following  the  directions  there  given  for  filling  the     cylinder, 
the  cylinder  being  connected  directly  to  the  source  of  the  gas 
used.     Note  the  time  of  efflux  between  the  same  two  positions 
for  the  float  as  used  in  (a).    This  insures  the  same  conditions 
as  to  pressure  in  the  two  cases. 

(c)  Repeat  (b),  using  dry  carbon  dioxide. 

(d)  Calculate  the  relative  densities,  referred  to  air,  of  the 
gases  used  in   (b)   and   (c).     Taking  the  density  of  dry  air 
under  standard  conditions  to  be  0.001293  gms.     per  cc.,  find 
the  density,  under  standard  conditions,  of  the  gases  used.  What 
"Laws"  have  been  used,  or  assumptions  made,  in  answering 
the  requirement  of  the  preceding  sentence? 

20.  ABSOLUTE  CALIBRATION  OF  A  THERMOM- 
ETER. 

References. — Watson's  Practical  Physics,  p.   162;  Edser,  p.  23.    . 

The  object  of  this  experiment  is  to  plot  a  curve  from  which 
the  true  temperature  may  be  obtained  corresponding  to  each 


46  ABSOLUTE    CALIBRATION    OF    A    THERMOMETER.  [2O 

scale-reading  of  a  given  mercurial  thermometer.  Such  a 
curve  is  called  the  calibration  curve  of  the  thermometer.  The 
process  of  obtaining  it  is  absolute  since  it  does  not  involve 
comparison  with  a  standard  thermometer. 

(a)  Correction  near  the    Lower    Fixed    Point. — Put  the 
thermometer  through  the  cork  in  a  test-tube,  having  rilled  the 
latter  about  half  full  of  distilled  water.     Place  the  tube  in  a 
freezing  mixture  of  shaved  ice  and  salt,  and  stir     the  water 
around  the  thermometer  until  it  begins  to  freeze.     Read  the 
thermometer.    By  warming  the  tube  in  the  hand  and  repeating 
the  freezing  process,  obtain  several  readings.     Let  us  suppose 
that  the  mean  of  these  readings  is  +  0.2 °C.     Since    the  true 
temperature  of  freezing  water  is  o°C.,  the  correction  corres- 
ponding to  the  given  scale-reading     of  the     thermometer     is 
—  0.2°,  for  this  when  added  to  the  reading  gives     the     true 
temperature. 

(b)  Correction  Near  the   Upper  Fixed  Point. — Place  the 
thermometer    through  the  cork     in  the  tube  at  the  top  of  the 
boiler,  with  the  bulb  well  above  the  surface  of     the     water. 
Boil  the  water  so  that  the  steam  issues  freely,  but  not  with  any 
perceptible  pressure,   from   the   upper  vent.     Read  the  ther- 
mometer when  it  becomes  steady.     Allow  the  boiler  to  cool 
slightly,  and  repeat,  making  three  readings  in  all.     If  the  in- 
strument be  provided  with  a  water-manometer,  take  the  man- 
ometer-reading simultaneously  with  the  temperature-reading. 
Read  the  barometer  and  determine  the  pressure  of  the  steam, 
and  find  from  the  Tables  the  true  boiling-point  temperature  for 
this  pressure.     Let  us  suppose  that  the  mean  of  the  readings 
of  the  thermometer  is  99.i°C.,  while  the  true  temperature  is 
99.8° C.    Then  the  correction  corresponding  to  this  scale-read- 
ing of  the  thermometer  is  +0.7°,  for  this  when  added  to  the 
reading  gives  the  true  temperature. 

(c)  Let  the  thermometer  cool  slowly  to  about  the  tempera- 
ture of  the  room,  and  repeat  (a).     If  the  freezing  point  ob- 


2O]  ABSOLUTE  CALIBRATION  OF  A  THERMOMETER.  47 

served  now  is  different  from  that  observed  in  (a),  use  the 
mean  of  the  two  values  in  the  calibration  that  follows.  As- 
suming the  temperature  of  freezing  water  to  be  o°C.,  write  the 
corrections  of  the  thermometer  for  the  scale-readings  observed 
in  (a)  and  (&).  Record  these  two  corrections  by  points  on 
coordinate  paper,  having  as  abscissae  the  scale-readings  of 
the  given  thermometer  from  o°  to  110°,  and  as  ordinates  the 
corresponding  corrections  in  tenths  of  a  degree  but  on  a  larger 
scale.  Corrections  should  be  plus  (  +  )  if  they  are  to  be  added 
to  the  observed  to  give  the  true  temperatures,  minus  ( — )  if 
they  are  to  be  subtracted.  Connect  these  two  points  by  a 
straight  line.  The  ordinate  of  this  straight  line  at  any  point 
gives  the  correction  of  the  thermometer  at  that  scale-reading 
on  the  assumption  that  the  bore  of  the  thermometer  is  uni- 
form throughout  the  whole  range.  In  general  this  assump- 
tion is  not  justified,  and  there  must  be  added  to  this  correc- 
tion at  each  point  another  correction  due  to  the  inequalities  of 
the  diameter  of  the  bore.  In  order  to  determine  this  latter 
correction,  it  will  be  necessary  to  calibrate  the  tube. 

(d)  Calibration  of  the  Tube. — Break  off  a  portion  of  the 
thread  of  mercury  about  ten  degrees  in  length.     Ask  for  as- 
sistance, if  necessary.     Place  the  lower  end  of  the  thread,  ap- 
proximately ten  degrees  long,  at  the  zero-point  of  the  scale 
and  read  the  position  of  the  upper  end  to  tenths  of  a    degree. 
Then  place  the  lower  end  at  10°  and  read  the  position  of  the 
upper  end.    Repeat  with  the  lower  end  at  the  successive  points 
20°,  30°,  40°,  etc.,  up  to  90° ;  then  come  down  again  with  up- 
per end  at  100°,  90°,  80°,  etc.,  reading  the  position     of  the 
lower  end  each  time. 

(e)  Record  the  observations  and  Calculations/ in     tabular 
form  in  six  columns  as  follows : 

1 i )  The  reading  of  the  lower  end  of  the  thread. 

(2)  The  corresponding  reading  of  the  upper  end. 

(3)  The  length  of  the  thread  in  each  position. 

(4)  The  mean  length  /  for  each  interval. 


48  ABSOLUTE  CALIBRATION  OF  A  THERMOMETER.  [2O 

By  the  mean  length  for  each  interval  is  meant  the  mean  of 
the  reading  over  a  certain  interval  going  up  (say  from  30  to 
40)  and  over  the  same  interval  (40  to  30)  coming  down.  Find 
the  mean  value  of  all  these  mean  lengths  throughout  the  whole 
range  and  record  this  as  the  mean  length  L  of  the  thread  for 
the  whole  range. 

(5)  The  correction,  L — /,  for  the  length  of  each  interval, 
that  is,  the  difference  between  the  mean  length  for  all  intervals 
and  the  observed  length  for  each  interval. 

(6)  The  correction  for  the  upper  end  of  each  interval.  This 
is  the  correction  for  the  lower  end  of  the    interval  plus  the 
correction  for  the  length  of  the  interval)  since  a  correction  at 
any  point  evidently  affects  all  points  above  this.       The  cor- 
rection thus  found  for  any  point  represents  the  magnitude  of 
the  inequalities  of  the  bore  up  to  that  point.    It  must  be  added 
to  the  observed  reading  for  that  point  to  give  the  correct  read- 
ing.    The  corrections  should  be  recorded  with  proper  signs 
(See  Watson's  Practical  Physics,  p.   168.) 

(/)  To  construct  a  final  table  of  corrections  it  is  neces- 
sary to  add,  algebraically,  the  corrections  found  in  (c)  and 
in  (e,  6).  This  can  best  be  done  by  plotting.  On  the  plat  made 
in  (c),  plot  points  whose  abscissae  are  10°,  20°,  30°,  etc.,  and 
whose  corresponding  ordinates  are  found  by  measuring  from 
the  slanting  line,  already  drawn,  distances  equal  to  the  corres- 
ponding corrections  found  in  (e,  6) — measuring  up  or  down 
from  this  line  according  as  the  corrections  are  plus  or  minus. 
The  smooth  curve,  which  should  now  be  drawn  through  these 
plotted  points,  is  the  calibration  curve  of  the  thermometer. 

What  are  the  temperatures  corresponding  to  readings  of  o°, 
25°>  5°°>  75°  and  100°  on  the  given  thermometer? 


21  ]  RELATIVE   CALIBRATION    OF   A   THERMOMETER.  49 

21.     RELATIVE  CALIBRATION  OF    A  THERMOM- 
ETER. 

Most  varieties  of  glass  expand  at  different  rates  at  differ- 
ent temperatures,  hence,  even  with  a  thermometer  whose  bore 
has  been  carefully  calibrated  by  some  such  method  as  given  in 
Exp.  20,  the  reading  can  be  relied  upon  only  within  certain 
limits.  After  having  obtained  a  thermometer  whose  calibra- 
tion curve  is  accurately  known,  so  that  it  may  be  taken  as  a 
"standard,"  the  most  convenient  method  of  calibrating  other 
thermometers  is  by  direct  comparison  with  the  standard, 
hence  the  name  "relative  calibration."  If  the  calibration 
curve  of  the  standard  thermometer  can  be  relied  upon,  all  ir- 
regularities of  any  other  thermometer  can  be  corrected. 

The  thermometer  to  be  calibrated  in  this  experiment  is  a 
50°  thermometer  reading  to  tenths  of  a  degree.  Tie  the  ther- 
mometer to  the  standard  thermometer  with  soft  cotton  twine, 
winding  it  between  the  stems  so  as  to  separate  them  slightly. 
Put  the  bulbs  nearly  opposite  each  other ;  and  see  that  cor- 
responding divisions  are  as  nearly  opposite  as  is  consistent  with 
this  condition.  Suspend  the  two  securely,  with  the  bulbs  in 
the  middle  of  a  kettle  of  water,  and  steady  the  stems  by  catch- 
ing them  loosely,  without  pressure,  in  a  clamp.  The  thermom- 
eters are  to  be  read  by  a  short-focus  telescope,  which  slides 
easily  on  the  vertical  rod  of  its  stand.  This  should  be  set 
with  its  object-glass  at  a  distance  of  about  50  cm.  from  the 
thermometers,  which  should  be  perpendicular  to  its  axis. 
When  taking  a  reading,  always  set  the  telescope  so  that  the 
top  of  the  mercury  column  appears  in  the  middle  of  the  field 
of  view  (not  near  its  upper  or  lower  edge)  in  order  to  avoid 
parallax. 

(a)  Take  a  careful  series  of  readings,  to  hundredths  of  a 
degree,  at  intervals  of  2°  or  3°  from  about  5°  to  45°.  Keep 
the  water  well  stirred,  and  keep  the  temperature  fairly  cori- 


5<D  VARIATION  OF  BOILING  POINT  WITH   PRESSURE.  [22 

stant  for  a  few  minutes  before  each  reading.  A  good  plan  is 
to  take  a  preliminary  reading  of  each  thermometer  in  order  to 
see  about  where  the  reading  is  going  to  come.  The  two  exact 
readings  can  then  be  made  so  quickly  as  to  be  practically  sim- 
ultaneous. Read  again  in  a  few  seconds,  taking  the  thermom- 
eters in  reverse  order.  Repeat,  if  necessary,  until  the  differ- 
ences obtained  for  two  such  readings  agree  fairly  well. 

(b)  Let  the  observers  change  places,  and  take     a  similar 
descending  series,  cooling  the  water  by  dipping  out  hot  and 
adding  cold  water. 

(c)  Ask  to  see  the  calibration  curve  of  the  standard  used, 
and  from  it  construct  a  table  of  corrections  for  the  thermom- 
eter you  are  calibrating.  Plot  a  calibration  curve,  recording  the 
number  of  the  thermometer.    In  your  future  work  with  a  ther- 
mometer of  this  type,  use  the  one  you  have  calibrated. 


22.       VARIATION     OF     BOILING     POINT     WITH 
PRESSURE. 

Reference. — Duff,    p.    229. 

There  are  two  methods  employed  in  studying  the  variation 
in  the  boiling  point  of  a  liquid  with  the  pressure  upon  its  free 
surface.  By  the  dynamic  method  the  pressure  above  the 
boiling  liquid  is  varied  by  means  of  an  air-pump  and  the  cor- 
responding temperature  observed.  By  the  static  method  the 
temperature  of  the  liquid,  suitably  enclosed,  is  varied  by 
means  of  baths  and  the  corresponding  pressure  observed. 
The  object  of  the  present  experiment  is  to  study  the  variation 
in  the  boiling  point  of  water,  employing  the  dynamic  method. 

The  apparatus  consists  of  an  air-tight  boiler  to  hold  the 
liquid,  a  steam-condenser,  around  which  cold  water  circulates, 
an  air-tight  chamber  large  enough  to  equalize  sudden  changes 
in  the  pressure,  an  air-pump  for  reducing  the  pressure  and  a 


22]  VARIATION  OF  BOILING  POINT  WITH  PRESSURE.  51 

manometer  for  measuring  the  same.  These  are  connected  up 
in  the  order  named  and  made  air-tight  so  far  as  the  air  out- 
side is  concerned. 

(a)  The  circulation  of  water  should  first  be  started  through 
the  steam-condenser,  which  is  a  glass  or  metal  tube  used  to 
jacket  the  tube  leading  from  the  boiling-flask,  thus  condensing 
the  steam  as  it  comes  from  the  flask.    The  thermometer  should 
be  passed  through  the  stopper  of  the  flask  and  so  regulated 
that  its  bulb  will  be  in  the     rising  steam,  and  not     in  the 
water.     The  connection  with  the  large  glass  bottle  serves  to 
equalize  sudden  changes  in  pressure  due  to  irregularities  in 
the  boiling.    In  heating  the  water  do  not  play  the  flame  on  the 
flask  directly  under  the  glass  beads,  but  rather  to  one  side  and 
below  the  water-line. 

First  boil  the  water  at  atmospheric  pressure,  reading  the 
manometer  and  noting  the  temperature.  Then  take  a  series  of 
readings  at  intervals  of  about  5  cm.  pressure,  until  the 
"bumping"  becomes  so  violent  as  to  render  further  readings 
impracticable.  Before  each  reading,  after  pumping  to  the  pres- 
sure desired,  close  the  stop-cock  over  the  jet-pump,  wait  a 
short  time  for  the  pressure  to  reach  equilibrium,  and  then 
make  the  reading  of  boiler  temperature  and  corresponding 
pressure.  Put  the  pump  again  in  connection,  obtain  a  new 
pressure,  and  repeat  the  readings.  Before  turning  off  the 
water  at  the  jet,  be  sure  each  time  to  let  air  into  the  apparatus 
by  opening  the  pinch-cock  nearest  the  pump,  otherwise  water 
will  flow  back  into  the  tubing. 

(b)  Take  a  series  of  readings  with     increasing     pressures 
up  to  atmospheric  pressure,  choosing  values  different  from  the 
previous  ones. 

(c)  Plot  the  observations  on  coordinate  paper,  using  pres- 
sures as  ordinates  and  temperatures  as  abscissae.     From  the 
curve  find  the  boiling  point  of  water  at  a    pressure  of  1/2 
atmosphere.    Discuss  the  phenomena  of    this    experiment    in 


52  COEFFICIENT  OF  EXPANSION   OF  A  LIQUID.  [23 

connection  with  the  difficulties  experienced  in  cooking  food  at 
high  altitudes.  Could  determinations  of  the  boiling  point  of 
water  be  used  to  measure  altitude,  and  how  ? 


23.     COEFFICIENT  OF  EXPANSION  OF  A  LIQUID 
BY  ARCHIMEDES'  PRINCIPLE. 

The  coefficient  of  expansion  of  a  heavy  oil  is  to  be  obtained 
by  observing  the  change  in  the  buoyant  force  acting  on  a  metal 
cylinder  when  immersed  in  the  oil  at  different  temperatures. 
A  brass  cylinder  is  suspended  from  one  arm  of  the  balance 
and  carefully  weighed,  first  in  air,  then  in  water  at  a  known 
temperature.  The  oil  is  then  placed  in  a  calorimeter  con- 
sisting of  one  beaker  inside  another,  and  the  cylinder  is 
weighed  when  immersed  in  the  oil,  the  temperature  of  the  oil 
being  noted,  which  should  be  the  same  as  that  of  the  water, 
or  nearly  so.  Since  the  oil  thickens  if  cooled,  it  is  convenient 
to  make  the  first  weighings  at  the  room  temperature. 

After  weighing  in  the  cool  oil,  the  inner  beaker  is  removed 
and  placed  in  a  water-bath  heated  to  60°  or  7<D°C.  Replacing 
the  beaker  with  the  heated  oil  in  the  calorimeter  beaker,  the 
cylinder  is  again  weighed  in  the  oil,  the  temperature  of  the  oil 
during  the  weighing  being  carefully  noted. 

Let  M  =  the  mass  balancing  the  cylinder  when  in  air, 
ml  =  the  mass  balancing  the  cylinder  when  in  water, 
m2  =  the  mass  balancing  the  cylinder  when  in  cool  oil, 
m3  =  the  mass  balancing  the  cylinder  when  in  hot  oil. 
t{  =  the  temperature  of  the  cool  oil  and  the  water,  and 
tz  =  the  temperature  of  the  hot  oil. 

Of  the  quantities  not  directly  measured,  but  which  must  be 
known  in  order  to  find  the  coefficient  of  expansion  of  the  oil, 
(i)  let  Vx  represent  the  volume  of  the  brass  cylinder  at  the 
lower  temperature  t,  and  d  the  density  of  the  water  at  the 


23]  COEFFICIENT  OF  EXPANSION  OF  A  LIQUID.  53 

same  temperature   (see  the  Tables).     By  Archimedes'  princi- 
ple and  the  definition  of  density 

__  M—  ml 

l~ 


(2)  Let  Vz  represent  the  volume  of  the  cylinder    at     the 
higher  temperature  t2  and  a  the  coefficient  of  cubical  expan- 
sion of  brass  (see  the  Tables).    By  definition  of  a,    •;«, 

F2  ==   V,    [I   +  a   (f,  -  O] 

(3)  Let  di  and  J2  represent  the  densities  of  the  oil  at  the 
lower  and  higher  temperatures  respectively.     By  Archimedes' 
principle  and  the  definition  of  density 

M  —  m*  M  —  m, 

a,  =       —  ~  —  and         a2  =  •  -  ~  —  . 

V  \  V   '2 

(4)  Finally,  let  ft  represent  the  coefficient  of  cubical  expan- 
sion of  the  oil.  If  any  mass  m  of  oil  has  a  volume  V  at  a 
temperature  ti  and  a  volume  V"  at  a  higher  temperature  J2> 
then  V"  =  V  [i  -f  ft  (t2  —  fj].  Dividing  both  members  by 
w,  and  substituting  for  m/V\  and  m/V"  their  equivalents  </± 
and  rf2,  the  equation  becomes  d^  =  dz  [i  +  /?  (£2  —  O]« 
This  is  true  generally  and  therefore  in  the  present  experiment. 
B  this  last  relation 


Carry  out  the  experiment  as  outlined  and  calculate  ft. 

Point  out  the  sources  of  error.  If  the  brass  cylinder  should 
have  had  an  internal  cavity,  show  what  its  effect  upon  the 
value  of  would  be. 


54    COMPARISON  OF  ALCHOHOL  AND  WATER  THERMOMETER.     [24 


24.     COMPARISON   OF   ALCOHOL  AND   WATER 
THERMOMETERS. 

In  this  experiment  the  relative  expansions  of  water  and 
alcohol  are  to  be  studied,  and  the  behavior  of  these  liquids 
when  used  in  thermometers  to  be  observed. 

(a)  Two  thermometer-bulbs  are  to  be  rilled,  one  with  water 
the  other  with  ethyl  alcohol,  by  the  aid  of  the  reservoir-tube. 
The  reservoir  is  fitted  on  the  end  of    the    thermometer-stem, 
filled  with  water   (or  alcohol),     and     warmed.     The     water 
should  first  be  boiled  to  drive  out  the  oxygen  held  in  solution, 
before  filling  the  reservoir  with  it.     The  liquid  is  then  intro- 
duced into  the  thermometer-bulb  by  alternately  heating  the 
bulb  to  drive  out  the  air  and  allowing  it  to  cool  to  admit  the 
liquid.     When  all  but  a  tiny  bubble  of  air  has  been  removed, 
place  the  bulb  in  ice-water  and  force  the  liquid  to  dissolve  the 
air.     If  this  does  not  succeed,  ask  for  assistance.     Take  care 
not  to  ignite  the  alcohol.     The  liquid  in  each     thermometer 
should  stand  i  or  2  cm.  above  the  lower  end  of  the  stem  when 
the  bulb  is  in  melting  ice. 

(b)  Glue  or  otherwise  fasten  a  strip  of  stiff  paper  along  the 
back  of  each  stem,  and  use  it  as  a  scale.    Then  place  the  ther- 
mometers in  clamps  with  their  bulbs  in  shaved  ice  or  in  a  mix- 
ture of  water  and  ice.    When  the  reading  becomes  steady,  indi- 
cate the  position  of  the  meniscus  of  each  by  a  sharp  line  on  the 
card.     Mark  the  line  zero.    This  is  the  first  fixed  point  of  the 
thermometer. 

(c)  To  determine  the  second  fixed  point,  place  the  bulbs 
in  a  beaker  of  wood  alcohol  which  is  itself  placed  on  a  sup- 
port in  a  bath  of  water.    Heat  the  water-bath  slowly  until  the 
wood  alcohol  begins  to  boil.    Be  very  careful  not  to  bring  the 
alcohol  itself  to  the  flame,  and  avoid  inhaling  the  fumes  of 
wood  alcohol.     When  steady,  again  indicate  the  position  of 


25]  COEFFICIENT  OF   EXPANSION   OF  A  LIQUID.  55 

the  meniscus  on  each  stem  by  a  sharp  line.  Mark  this  point 
66,  which  is  the  boiling-  point  of  wood  alcohol  on  the  centi- 
grade scale. 

(d)  Lay  the  stems  of  the  thermometers  on  a  flat  surface, 
measure  the  distance  on  each  between  the  two  fixed     points, 
and  divide  this  distance  into  66  equal  parts,  calling  each  part 
a  degree.     Put  the  marks  for  each  degree  on  the  scale  and 
number  every  tenth  one. 

(e)  Place  the  two  arbitrarily  calibrated  thermometers  in 
a  water-bath  at  o°,  as  recorded  by  each  thermometer.    Gradu- 
ally raise  the  temperature  of  the  water-bath  and  note  the  read- 
ings of  the  two  thermometers,  at  first  at  short  intervals,  then 
at  longer  intervals,  until  the  upper  fixed     point     is     reached. 
Each  time  that  the  temperature  is  raised,  it  should  be  kept  at 
its  new  value  for  three  or  four  minutes,     so  as  to  give  the 
bulbs  time  to  assume  the  temperature  of  the  bath. 

(f)  Plot  the  series  of  points  on  coordinate  paper,  having  as 
abscissae  the  temperatures  by  the  alcohol-thermometer,  and  as 
ordinates  the  corresponding     temperatures     by     the     water- 
thermometer.    Draw  a  smooth  curve  through  the  points.  This 
curve  gives  the  relation  between  the  temperatures  as  recorded 
by  the  two  thermometers.     What  inferences  can  you  draw 
from  the  curve  ?    If  alcohol  be  taken  as  the  standard  substance, 
what  can  you  say  of  the  uniformity  of  the  expansion  of  the 
water?    If  the  water  be  assumed  as  the  standard,  what  of  the 
expansion  of  the  alcohol?     Which  would  be  the  better  sub- 
stance to  use  in  a  practical  thermometer,  and  why? 

25.     COEFFICIENT  OF  EXPANSION  OF  A  LIQUID 
BY  REGNAULT'S  METHOD. 

References. — Duff,  p.  199;  Edser,  pp.  71,  76. 

This  method  was  originally  devised  by  Dulong  and  Petit, 
but  was  improved  and  made  practical  by  Regnault.     It  is  an 


56 


COEFFICIENT    OF    EXPANSION    OF    A    LIQUID. 


absolute  method  in  that  the  effect  of  the  expansion  of  the  con- 
taining vessel  is  eliminated.  It  is  applicable  to  any  liquid.  The 
purpose  of  the  present  experiment  is  to  determine  the  coeffi- 
cient of  expansion  of  turpentine  or  olive  oil  by  means  of  this 
method. 

Two  glass  tubes,  A  A'  and  B  B',  are  surrounded  by  metal 
cylinders,  L  and  M  respectively,  in  which  baths  at  different 
temperatures  may  be  placed.  These  glass  tubes  are  connected 
near  the  top  by  a  horizontal  tube  A  C  B  from 
which  there  extends  an  upright  open  tube  C,  and  are  connected 
at  the  bottom  between  A'  and  B'  by  an  inverted  U-tube.  The 
liquid  is  poured  into  the  glass  tubes  until  it  stands  at  some  point 
in  C  just  above  the  horizontal  level  AB.  This  insures  the 
height  remaining  the  same,  or  very  nearly  the  same,  at  A  and 
B,  and  gives  a  means  of  measuring  the  height  without  observ- 
ing the  meniscus  at  A  or  B.  In  the  bend  of  the  tube  GK 
there  is  compressed  air,  so  that  the  pressure  is  always  the  same 
at  the  meniscus  G  and  the  meniscus  K.  The  levels  G,  K,  and 
C  may  be  measured.  Cold  water  is  passed  through  M,  or  a 
mixture  of  ice  and  water  placed  in  it,  and  steam  is  passed 
through  L,  thermometers  at  A  and  B  indicating  the  tempera- 


25]  COEFFICIENT   OF   EXPANSION    OF    A    LIQUID.  57 

tures.  Suppose  that  the  temperature  of  M  is  /1°C.,  of  L  is 
tz°C.,  and  of  the  tubes  between  them  is  J8°C.  or  the  fefflpera=- 
ature  of  the  room.  Let  d^  be  the  density  of  the  cold  liquid  in 
M  Sit  /!0C. ;  d2  the  density  of  the  hot  liquid  in  L  at  f8°C. ;  c?, 
the  density  of  the  liiquid  in  the  tubes  between  M  and  L  at 
*3°C.;  H  the  vertical  height  of  C  above  the  level  of  A'B' \  ?, 
and  /2  the  lengths  of  the  columns  KD  and  GB,  and  h  their  dif- 
ference GK.  Then,  since  the  pressure  of  the  air  enclosed  in 
the  U-tube  is  the  same  at  G  as  at  K,  we  have 

(i)      p  +  H.  d2  g  —  12  d3  g  =  p  +  H  d,  g  —  I,  d3  g, 
where  p  is  the  atmospheric  pressure  and  g  is  the  acceleration 
due  to  gravity. 

If  ft  is  the  coefficient  of  cubical  expansion  of  the  liquid, 
referred  to  the  volume  at  o°C.,  d0  its  density  at  o°C.,  V \  the 
volume  of  a  given  mass  m  of  it  at  o°C.,  and  V ^  the  volume  of 
the  same  mass  at  ^°C.,  we  have  V^  =  VQ  (i  -(-  /?  f±). 
Since  VQ  =  m/d0  and  V1  =  m/dlt  it  follows  by  substitution 
that  d0  =  dl  (i  +  0  fj.  This  is  true  generally,  and  there- 
fore in  the  present  experiment.  Hence 

(2)  d0  =  d,  (i  +  /?  fx), 

(3)  ^o  =  ^2  (i  +^^2), 

(4)  ^0  =  ^3  (i  +j8*,)- 

Substituting  in  (i)  the  values  of  rf^  d2  and  J3  obtained  from 
(2),   (3)  and  (4)  and  simplifying,  we  get 

H  h  H 

i  H-  y?4  +  i  -f  #,      i  +  fa 

Clearing  of  fractions  and  putting  the  equation  into  the  form 
a/?2  -|-  bp  +  c  =  o?  we  get  by  solution 


where  a  =  H  tt  t3  +  h  tt  t2  —  H  t2  ts, 

b=-H  (ti  +  tj  +h  (t,  +  t2)—H  (tz  +  #,),  and 

c  -  +  A, 


58  COEFFICIENT  OF  EXPANSION  OF  GLASS.  [26 

By  measuring  the  lengths  H,  h  and  the  three  temperatures  tit 
/2,  £3,  the  value  of  ft  may  at  once  be  calculated. 

(a)  Pass  cold  running  water  through  the  jacket  M  and 
steam  through  L,  keeping  thermometers   on  the  two     sides. 
Instead  of  running  water,  it  may  be  necessary    to  use  a    bath 
of  ice  and  water  in  M.     Wrap  a  cloth  about  the  horizontal 
tube  A'E,  and  keep  this  wet  with  cold  water  to  make  the  con- 
duction of  heat  to  the  column  GH  as  small  as  possible.     Note 
the  difference  in  height  of  the  menisci  at  G  and  K    when    the 
temperatures  become  steady.     The  distance  h  must  be     very 
accurately  determined  at  each  setting,  whereas  a  moderately 
accurate  reading  of  the  height  H  is  sufficient.     Do  not  forget 
to  take  simultaneous  readings  of  the  heights  and  the  tempera- 
tures.   Continue  the  experiment  long  enough  to  be  certain  that 
the  conditions  are  steady. 

(b)  Calculate  the  value  of  /?.    Why  should    this    often  be 
called  the  mean  zero  coefficient  of  the  liquid  between  o°  and 
the  temperature  of  the  steam? 

If  the  height  h  were  measured  accurately  to  within  o.i  mm. 
to  what  accuracy  should  the  height  H  be  measured? 

26.     COEFFICIENT  OF  EXPANSION  OF  GLASS  BY 
WEIGHT-THERMOMETER. 

The  purpose  of  this  experiment  is  to  determine  the  coeffi- 
cient of  cubical  expansion  of  glass  by  means  of  the  weight- 
thermometer. 

The  weight-thermometer  consists  of  a  glass  tube  closed  at 
one  end  and  ending  in  a  curved  capillary  at  the  other  end.  It 
is  filled  with  mercury  at  o°C.,  and  the  mass  of  the  mercury 
measured.  When  later  placed  in  a  bath  of  higher  tempera- 
ture, some  mercury  overflows,  since  mercury  expands  more 
rapidly  when  heated  than  does  glass.  The  mass  of  this  over- 
flow is  measured. 


26]  COEFFICIENT    OF    EXPANSION    OF    GLASS.  59 

Let  M  =  the  mass  of  the  mercury  filling  the  weight-ther- 
mometer at  o°C, 

j/o  =  the  volume  of  M,  and  hence  of  the  weight-thermom- 
eter at  o°C., 

ft,  y  =  the  coefficients  of  cubical  expansion  respectively  of 
mercury  and  glass,  and 

m  =  the  mass  of  the  mercury  which  overflows  when  the 
temperature  is  raised  from  o°  to  t°C. 
Then    VQ  (i  +  fit)  =  the  volume  of  the  mass  M  at  t°, 
and  VQ  (i  +  yO  =  the  volume  of  the  weight-thermometer  at 
f°;  hence  VQ  (i  +  pt)  -  -  V,  (i  +  yt)  =  F0(0  —  y)  t  — 
the  volume  of  the  mass  m  at  t°.     If  d0  and  dt  represent  the 
densities  of  mercury  at  o°  and  t°,  respectively,  then 

(1)  d0  =  M/V0, 

(2)  dt  =  m/V0  (13  —  y)  t, 

(3)  d0  =  dt(i+00       (SeeExp.  25.) 

By  eliminating  d0  and  dt  from  equations  (i),  (2),  and  (3), 
we  get 


Besides  containing  the  known  masses,  M  and  m,  this  equation 
contains  the  three  quantities,  ft,  y,  and  /.  Any  two  of  these 
three  quantities  being  known,  the  third  will  be  given  by  the 
equation.  In  the  present  experiment  ft  and  t  are  known,  and 
y  is  to  be  calculated. 

(a)  Weigh  the  empty  weight-thermometer  to  an  accuracy 
of  10  mg.  Then  fill  it  with  mercury.  In  doing  so  it  should 
be  held  by  a  clamp,  or  suspended  in  a  gauze  jacket,  and  heated 
by  a  flame  held  in  the  hand,  care  being  taken  to  keep  from 
heating  too  rapidly  and  from  applying  the  flame  too  long  at 
any  point  of  the  empty  bulb. 

The  end  of  the  capillary  dips  under  the  surface  of  mercury 
in  a  porcelain  dish.  The  mercury  in  this  dish  should  first  be 


60  COEFFICIENT  OF  EXPANSION   OF  GLASS.  [26 

heated,  and  then  the  weight-thermometer  gently  heated  untfl 
the  air  bubbles  out  through  the  mercury.  On  allowing  the 
bulb  to  cool,  some  mercury  will  run  into  it.  The  process  is 
then  repeated.  When  considerable  mercury  is  in  the  bulb, 
heat  it  until  it  boils  vigorously,  but  be  careful  not  to  heat  too 
hot  that  portion  of  the  glass  where  there  is  no  mercury.  Keep 
the  mercury  hot  in  the  porcelain  dish,  otherwise  the  glass  is  apt 
to  crack  when  the  cooler  mercury  rushes  in.  The  tube  must 
be  completely  filled  with  mercury  to  the  end  of  the  capillary, 
the  last  bubble  of  air  being  expelled.  To  accomplish  this  it 
will  be  found  helpful  to  turn  the  weight-thermometer  so  as  to 
give  the  capillary  above  the  air-bubble  an  upward  slant.  A 
gentle  tapping  with  a  light  splinter  or  pencil  will  then  prob- 
ably cause  the  bubble  to  work  its  way  along  the  tube  far 
enough  to  be  easily  expelled  by  further  heating. 

(6)  Keeping  the  end  of  the  capillary  in  the  dish  of  mer- 
cury, allow  the  weight-thermometer  to  cool  in  the  air  suffi- 
ciently so  that  you  can  bear  your  hand  on  it.  Then  surround 
it  with  shaved  ice  and  leave  it  long  enough  to  contract  as  much 
as  it  will.  Assume  that  its  temperature  is  now  o°C.  Carefully 
remove  the  dish  and  brush  the  mercury  off  the  end  of  the  cap- 
illary. Place  a  watch-glass  under  the  end  to  catch  the  mercurv 
as  it  begins  to  expand  and  flow  out.  Now  remove  the  ice- 
bath  and  warm  the  bulb  with  the  hand  until  its  temperature 
is  raised  to  the  temperature  of  the  room. 

(c)  Place  the  weight-thermometer  in  the  boiler  provided. 
Heat  it  to  the  boiling  point  of  water  by  passing  steam  over  it 
until  no  more  mercury  comes  out.     Read  the  barometer  and 
calculate  the  temperature  of  the  steam.     Very  carefully  weigh 
the  mercury  in  the  watch-glass  to  an  accuracy  of  i  mg.  Weigh 
the  weight-thermometer  and  contained  mercury  to  an  accuracy 
of  10  mg. 

(d)  Using  your  values  of  M  and  m,  and  taking  the  coeffi- 
cient of  expansion  of  mercury  as  found  in  Exp.  25,  or  from 


27]  COEFFICIENT  OF  EXPANSION  OF  A  LIQUID.  6l 

the  Tables,  calculate  the  coefficient  of  cubical   expansion   of 
glass. 

What  additional  measurements  would  you  need  to  have 
made  in  order  to  measure  the  room  temperature  with  your 
weight-thermometer  ? 


27.     COEFFICIENT  OF  EXPANSION  OF  A  LIQUID 
BY  PYCNOMETER. 

Reference. — Edser,    p.    81    and    p.    86. 

The  method  consists  in  determining  the  mass  of  the  liquid 
(alcohol)  filling  a  pycnometer  at  each  of  several  different  tem- 
peratures, and  from  the  data  calculating  the  coefficient  of  ex- 
pansion of  the  liquid.  Four  determinations  should  be  made, 
at  intervals  of  about  8°,  beginning  with  the  room  tempera- 
ture. 

(a)  Fill  the  pycnometer  with  alcohol  and  set  it  on  a  plat- 
form in  a  kettle  of  water,  so  that  the  water  comes  well  up  to 
the  neck  of  the  pycnometer.     Hang  a  50°  thermometer  in  the 
bath  alongside  the  pycnometer,  and  keep  the  bath  well  stirred 
for  about  five  minutes.     The  temperature  of  the  bath,  which 
must  be  a  little  above  that  of  the  room,  should  remain  con- 
stant within  o°.i  during  this  time,  and  at  the  end  of  it  the 
alcohol  will  have  the  same  temperature     within     o°.i.     Take 
the  pycnometer  out  of  the     bath,  wipe     the  outside  dry,  and 
weigh  to  an  accuracy  of  i  mg. 

(b)  Repeat  with  the  bath  at  or  near  each  of  the  higher  tem- 
peratures  selected,  keeping  the   temperature   steady   for     ten 
minutes  by  holding  the  lamp  under  the  kettle  for  a  few  sec- 
onds occasionally.     Careful  trial  has  shown  that  after     this 
treatment  the  temperature  of  the  alcohol  at  the  center  of  the 
pycnometer  is  about  o°.i  lower  than     that  of    the  bath,  and 
therefore  the  average  temperature  of  the  alcohol  is  the  same 
as  that  of  the  bath  to  within  o°.i. 


62  EXPANSION  CURVE  OF  WATER.  [28 

(c)  Empty  the  alcohol  into  the  bottle  from  which  it  was 
taken,  dry  the  pycnometer  with  a  jet-pump,  and  weigh. 

(d)  Determine  the  mass  of  the  alcohol  filling  the  pycnometer 
at  each  temperature.     Plot  the  results,     with     temperatures, 
starting  from  o°,  as  abscissae,  and  masses  as  ordinates.  As- 
suming that  the  expansion  is  uniform,  draw  the  straight  line 
which  best  represents  the  plotted  points,  and  from  it  find  the 
mass  filling  the  pycnometer  at  o°C. 

(e)  For  the  time  being,  assume  that  the  volume  of  the 
pycnometer  remains  constant.    Then  the  ratio  of  the  masses  of 
alcohol  filling  this  volume  at  any  two  temperatures   (o°  and 
40°  C.,  say)  is  equal  to  the  inverse  ratio  of  the  volumes  of  a 
given  constant  mass  at  the     given     temperatures.     (Proof?) 
From  this  calculate  the  apparent  coefficient  of     expansion  of 
alcohol  for  the  given  range  of  temperature. 

(/)  What  has  been  found  is  not  the  absolute  coefficient  of 
expansion,  since  the  pycnometer  also  expands.  Find  from 
your  own  work  in  Exp.  26,  or  from  the  Tables,  the  coefficient 
of  cubical  expansion  of  glass,  and  by  applying  it  to  the  above 
result  find  the  absolute  coefficient  for  the  alcohol. 

(g)  The  formula  derived  in  Exp.  26  for  the  weight-ther- 
mometer will  also  apply  to  the  pycnometer  as  used  in  the  pres- 
ent experiment.  By  means  of  this  formula  determine  the  co- 
efficient of  expansion  for  alcohol.  Is  the  coefficient,  as  thus 
determined,  apparent  or  absolute? 

28.   EXPANSION    CURVE   OF   WATER. 

Reference. — Duff,    p.    201. 

The  variation  of  the  volume  of  a  given  mass  of  water,  as 
the  temperature  is  raised  by  steps  from  the  freezing  point,  is 
to  be  studied,  taking  the  expansion  of  mercury  as  the  tem- 
perature standard.  It  should  be  remembered  that  our  choice 
of  a  thermometer  and  scale  of  temperatures  is  entirely  arbi- 


28]  EXPANSION  CURVE  OF  WATER.  63 

trary.  The  statement  that  a  certain  substance  expands  "uni- 
formly'' can  mean  only  that  it  expands  uniformly  with  the 
change  of  some  property  of  a  particular  substance  chosen  as 
a  standard.  Taking  the  expansion  of  mercury  as  a  standard, 
we  wish  here  to  determine  how  water  changes  in  volume  with 
change  of  temperature. 

(a)  The  bulb  of  the  water-thermometer  can  be  filled  by  the 
aid  of  a  reservoir-tube  fitted  on  the  end  of  the  thermometer- 
stem.     The  reservoir  is  filled  with  water  previously  boiled  to 
expel  the  oxygen  dissolved  in  it.     The  bulb  is  then  alternately 
heated  to  drive  out  the  air  and  allowed  to  cool  to  admit  water. 
Finally,  when  only  a  tiny  air-bubble  remains,  this  may  be  got- 
ten rid  of,  either  by  jarring  the  tube  so  as  to  break  up  the 
bubble  into  smaller  ones  which  will  pass  up  along  the  stem,  or 
by  immersing  the  bulb  in  ice-water  and  forcing  the  air  back 
into  solution.     I'f  neither  of  these  methods  works,  ask  for  as- 
sistance. 

Fill  the  thermometer  until  the  water  stands  in  the  stem,  at 
o°C.,  about  2  cm.  above  the  bulb.  Fasten  the  tube  to  the  face 
of  a  metric  scale  and  place  the  bulb  in  the  water-bath.  The 
bulb  is  first  to  be  surrounded  with  shaved  ice.  When  condi- 
tions become  constant,  take  a  reading  of  the  height  of  the 
water  meniscus  and  also  of  the  mercury  thermometer  placed 
in  the  bath  near  the  bulb.  Melt  the  ice  and  gradually  raise 
the  temperature  of  the  bath  very  carefully,  at  first  reading  the 
mercury  and  water  thermometers  at  every  degree  between  o° 
and  8°C.,  then  at  approximately  10°,  15°,  20°,  and  every  ten 
degrees  thereafter  as  far  as  the  water  thermometer  will  per- 
mit. 

(b)  Determine  the  volume  of  the  water  in  the  bulb  at  o° 
C.,  by  weighing  the  bulb  with  the  water  in  it  and  then  weigh- 
ing it  empty  and  dry. 

(c)  Determine  the  diameter  of  the  bore,  either  by     direct, 
measurement  with  the  micrometer  microscope,  or  by  placing 


64  EXPANSION   CURVE  OF  WATER. 

in  the  tube  a  thread  of  mercury,   measuring  its   length   and 
then  weighing  the  mercury. 

(d)  From  the  determination  of  the  volume  of  the  bull) 
and  the  diameter  of  the  bore  of  the  tube,  calculate  the  volume, 
in  cu.  cm.,  of  the  water  at  each  of  the  temperatures  observed, 
making  no  allowance  for  the  expansion  of  the  glass. 

On  coordinate  paper  plot  the  results  and  draw  a  curve,  hav- 
ing for  abscissae  the  temperatures  as  recorded  by  the  mercury 
thermometer,  and  for  ordinates  the  corresponding  volumes  of 
water.  In  doing  this,  choose  as  large  a  scale  for  volumes  as 
possible,  so  that  the  total  change  of  volume  will  about  cover 
the  width  of  the  sheet  of  paper.  In  order  properly  to  show 
the  expansion  between  o°  and  8°,  reproduce  this  part  of  the 
curve,  employing  a  magnified  volume-unit.  Since  your  ob- 
served changes  of  volume  are  only  apparent  changes,  the  vol- 
ume-change of  glass  must  be  added  in  order  to  obtain  the  true 
expansion  of  the  water.  To  do  this,  calculate  what  the  volume- 
increase  of  the  water-thermometer  is  between  o°  and  100°  C., 
due  to  the  expansion  of  the  glass,  using  the  coefficient  of 
cubical  expansion  of  glass  given  in  the  Tables.  At  the  point 
on  the  temperature  axis  corresponding  to  100°  erect  an  ordi- 
nate  equal  to  this  expansion.  Draw  a  slanting  line  through 
your  origin  of  coordinates  and  the  upper  end  of  this  ordinate 
The  lengths  of  the  ordinates  between  this  line  and  the  tempera- 
ture axis  represent  the  expansion  of  the  glass  for  the  corre- 
sponding temperatures.  Then,  from  various  points  along  the 
apparent  expansion  curve  of  water,  measure,  vertically  up- 
ward, distances  equal  to  the  volume-increase  of  the  glass  cor- 
responding to  this  temperature.  Draw  a  smooth  curve  through 
all  of:  the  points  thus  plotted.  This  curve  referred  to  the  hor- 
izontal axis  will  give  the  true  expansion  of  the  water. 

(e)     State  any  conclusions  that  can  be  drawn,  from  an  ex 
amination  of  the  curve,  in  regard  to  the  behavior  of  water  as 
its  temperature  is  raised  from  o°  to  the  highest  point  reached. 


28]  EXPANSION  CURVE  OF  WATER.  65 

By  the  use  of  the  curve  determine  the  mean  coefficient  of 
expansion  of  water  (i)  between  o°  and  100°,  (2)  between 
o°  and  20°,  (3)  between  o°  and  8°. 

Water  Equivalent. 

In  most  experiments,  such  as  those  involving  specific  heat, 
heat  of  fusion,  etc.,  where  a  calorimeter  and  its  accessories 
are  used,  it  is  convenient  to  know  their  water-equivalent. 

By  the  'water -equivalent  of  a  body  is  meant  the  number  of 
grams  of  water  which  would  be  heated  (or  cooled)  the  same 
number  of  degrees  as  the  body  for  the  passage  into  it  (or  out 
of  it)  of  the  same  amount  of  heat.  It  is  numerically  equal 
to  the  heat  capacity  of  the  body,  and  is  found  by  taking  the 
product  of  the  mass  of  the  body  and  the  specific  heat  of  the 
substance  of  which  it  is  made.  It  is  called  " water-equivalent" 
for  the  reason  that  in  all  calorimetric  calculations  the  body 
may  be  replaced  by  this  thermally  equivalent  mass  of  water. 
This  applies  directly  to  calorimeter  cups  and  stirrers.  As  a 
rule,  it  will  be  necessary  to  consult  the  Tables  for  the  values 
of  the  specific  heat. 

In  the  case  of  a  thermometer,  which  is  part  glass  and  part 
mercury,  the  water-equivalent  may  be  determined  by  finding 
the  volume  of  the  immersed  part  of  the  thermometer  and  then 
calculating  the  water-equivalent  of  this  volume  of  mercury. 
This  is  possible  since  equal  volumes  of  glass  and  mercury 
have  practically  the  same  heat  capacity,  and  hence  the  ther- 
mometer may  be  treated  as  though  it  were  made  entirely  of 
mercury.  The  student  will  find  that  0.45  is  approximately 
the  factor  by  which  the  volume  in  cc.  should  be  multiplied  to 
give  the  water-equivalent  of  the  thermometer. 


66      SPECIFIC   HEAT  OF  A   LIQUID  BY   METHOD  OF  HEATING.       |  2O 


29.     SPECIFIC  HEAT  OF  A  LIQUID   BY   METHOD 
OF  HEATING. 

In  this  experiment  a  heating  coil,  composed  of  high  resist- 
ance metal  through  which  an  electric  current  is  passed,  is 
immersed  for  a  given  time,  first  in  one  liquid  and  then  in 
another.  If  the  same  current  passes  through  the  coil  in  the 
two  cases,  equal  quantities  of  heat  should  be  generated  in 
equal  times.  Noting  in  each  case  the  mass  of  the  liquid  and 
the  rise  in  temperature,  the  two  quantities  of  heat  may  be 
equated  and  the  specific  heat  of  one  liquid  calculated,  if  that 
of  the  other  is  known.  Let  mlt  m2  be  the  masses  of  the  two 
liquids;  slt  s2  their  specific  heats;  f,,  t2  their  changes  in  tern 
erature ;  and  u*  the  water-equivalent  of  the  calorimeter  cup 
and  accessories.  Then 

(mi  S-L  +  w  0  *i  =  (m2  S2  +  w  i)  t2, 

from  which  s2  may  be  found  if  the  rest  of  the  quantities  are 
known.  Water,  taken  as  a  standard,  will  be  one  liquid  used. 
Another  liquid  is  furnished,  whose  specific  heat  it  is  the  object 
of  this  experiment  to  determine.  The  method  is  applicable 
to  any  liquid  which  is  not  a  conductor  of  electricity  and  which 
does  not  act  chemically  upon  the  material  of  the  coil  or 
calorimeter. 

(a)  Place  the  bottle,  containing  the  second  liquid,  in  a 
vessel  of  ice-water  to  cool.  Weigh  a  quantity  of  ice-water  in 
the  calorimeter  cup.  Set  up  the  calorimeter  and  immerse 
the  heating  coil,  having  the  temperature  of  the  water  about 
12°  below  the  room  temperature.  Allow  a  few  moments  for 
the  contents  of  the  cup  to  come  to  a  uniform  temperature, 
then  note  the  temperature,  and  turn  on  the  current  in  the  coil. 
Record  the  time  when  the  current  is  started,  and  also  the  time 
for  each  rise  of  two  or  three  degrees  in  temperature  of  the 
water  until  it  reaches  a  temperature  as  far  above  that  of  the 


30]      SPECIFIC   HEAT  OF  A  LIQUID  BY   METHOD  OF  COOLING.      67 

room  as  it  started  below.    Keep  the  water  well  stirred,  and  do 
not  place  the  thermometer  very  close  to  the  heating  coil. 

(b)  Repeat  (a),  using  the  second  liquid,  instead  of  water, 
in  the  calorimeter  cup.     It  will  be  necessary  to  find  the  water- 
equivalent  of  the  calorimeter  cup,    stirrer,  and    thermometer. 
For  this  purpose  see  the  paragraph  on  "Water-Equivalent." 

(c)  Plot  on  the  same  sheet  of  coordinate  paper  the  results 
of  (a)  and  (b),  using  temperatures  as  ordinates  and  times  as 
abscissae.     Erect  two  perpendiculars  to  the  time  axis  which 
will  include  between  them  as  wide  segments  of  the  two  curves 
as  is  consistent  with  accuracy.     From  these  intersections  ob- 
tain the  range  of  temperatures  passed  through  by  the  water 
and  the  other  liquid  in  equal  times.     Take  the  quantities  of 
heat  gained  by  the  two  liquids  and  the  calorimeter  in  this  time 
as  equal,  and  form  an  equation  from  which  the  specific  heat 
of  the  liquid  may  be  calculated.    From  the  result  just  obtained 
calculate  what  mass  of  the  liquid  will  be  "equivalent"  to  the 
water  used  in  (a). 

(d)  If  you  have  time,  take  the  amount  of  liquid  found  in 
(c)  to  be  equivalent  to  the  water  used  in  (a),  and  repeat  (b}. 
From  the  data  obtained  calculate  the  value  of  the  specific  heat. 
Why  should  this  value  be  more  reliable  than  the  one  found  in 

(0? 

30.     SPECIFIC  HEAT  OF  A  LIQUID  BY  METHOD 
OF  COOLING. 

Reference. — Millikan,    p.    206. 

This  method  is  a  comparison  of  the  quantities  of  heat  lost 
by  two  liquids,  one  of  which  is  water,  when  equal  volumes 
are  allowed  to  cool  under  exactly  the  same  conditions  through 
a  certain  range  of  temperature.  The  conditions  of  radiation 
being  the  same  for  both,  if  a  liquid  of  mass  m^  and  specific 
heat  s^  cools  through  a  certain  temperature-range  t  in  Tl  sec- 


68      SPECIFIC  HEAT  OF  A  LIQUID  BY   METHOD  OF  COOLING.       [30 

onds,  and  a  second  liquid  of  mass  w2  and  specific  heat  s2 
requires  T2  seconds  for  the  same  temperature-change,  then 
the  quantities  of  heat  lost  will  be  proportional  to  the  times, 
i-  e.,  QJQ2  =  7VT2.  If  w  denote  the  water-equivalent  of 
the  containing  vessel,  thermometer,  and  stirrer,  the  above  re- 
lation becomes 

(m,s,   +   w  i  )  t  ==  T, 
(  m,  s,  -f   w  i  )  /    ~  T.  ' 
from  which  the  unknown  specific  heat  can  be  determined. 

(a)  A  large  jar  is  used,  having  a  wooden  cover  from  which 
is  suspended  a  smaller  vessel.  The  space  between  these  is 
made  a  water-jacket  by  putting  enough  water  in  the  larger 
so  that  when  the  cover  is  put  in  place  the  space  between  the 
two  vessels  will  be  filled  with  water.  The  liquid  used,  tur- 
pentine, is  now  heated  in  a  water-bath  to  about  85°  and  poured 
into  a  small  copper  cup,  closed  by  a  cork  through  which  the 
thermometer  and  stirrer  pass.  The  cup  is  then  passed  through 
the  wooden  cover  and  hangs  suspended  from  the  cork. 

Make  the  necessary  weighings  on  the  trip-scales.  Allow 
the  turpentine  to  cool  to  about  50°,  recording  the  time  for 
every  two  degrees  fall  in  temperature  at  first,  and  later  for 
each  one  degree  fall. 

(&)  Put  fresh  water  in  the  jacket,  and  repeat  (a)  with 
water  in  the  cup  instead  of  turpentine,  using  as  nearly  as 
possible  the  same  volume. 

(c)  Plot  the  cooling  curves  of  turpentine  and  of  water  on 
coordinate  paper,  using  temperatures  as  ordinates  and  cor- 
responding times  as  abscissae.  On  these  curves  take  a  cer- 
tain range  of  temperature  by  drawing  two  lines  parallel  to 
the  axis  of  abscissae,  each  line  cutting  both  curves.  Make 
this  temperature-interval  as  long  as  possible,  consistent  with 
accuracy.  The  intersections  of  these  lines  with  the  curves  will 
give  the  times  required.  Calculate  the  specific  heat  of  tur- 
pentine. 


31  ]  MECHANICAL  EQUIVALENT  OF  HEAT.  6Q 

If  the  water  had  not  been  changed  between  the  two  sets 
of  observations,  in  what  way  would  the  value  for  the  specific 
heat  of  turpentine  have  been  affected?  What  source  of  error 
still  remains  even  if  the  water  in  the  jacket  is  changed  before 
the  second  measurement?  Suggest  a  way  to  avoid  this  un- 
certaintv. 


31.    MECHANICAL     EQUIVALENT   OF     HEAT   BY 
CALLENDAR'S  METHOD. 

The  number  of  units  of  mechanical  work  which  is  equiva- 
lent to  the  calorie  of  heat  is  called  the  mechanical  equivalent 
of  heat.  Most  of  the  methods  employed  in  determining  it 
produce  the  heat  by  means  of  mechanical  work  done  against 
friction.  In  the  Callendar  method  a  measurable  amount  of 
work  done  against  the  friction  between  a  stationary  silk  belt 
and  a  revolving  vessel  is  converted  into  heat  in  a  known  mass 
of  water  contained  in  the  vessel.  The  apparatus  consists  of  a 
brass  cylindrical  vessel  which  contains  a  known  mass  of  water 
and  whose  axis  is  horizontal.  This  cylinder  can  be  rotated 
at  a  moderate  speed  by  hand  or  by  motor.  Over  the  surface 
of  the  cylinder  a  silk  belt  is  wound  so  as  to  make  one  and  a 
half  complete  turns.  From  the  ends  of  this  belt  are  suspended 
known  masses,  adjusted  so  as  to  provide  a  force-moment 
which  will  oppose  the  rotation  of  the  vessel.  An  automatic 
adjustment  for  equilibrium  is  secured  by  the  use  of  a  light 
spring  balance  which  acts  in  direct  opposition  to  the  weight 
at  the  lighter  end  of  the  belt.  This  spring  balance  contributes 
only  a  small  part  to  the  effective  difference  of  load  between 
the  two  ends  of  the  belt,  hence  small  errors  in  its  reading 
are  relatively  unimportant.  The  masses  suspended  from  the 
belt  are  approximately  adjusted  by  trial  to  suit  the  friction  of 
the  belt?  the  final  adjustment  being  automatically  effected  by 
the  spring  balance.  A  counter  registers  the  number  of  revo- 


7O  MECHANICAL  EQUIVALENT  OF  HEAT.  [31 

lutions ;  and  a  bent  thermometer,  inserted  through  a  central 
opening  in  the  front  end  of  the  cylinder,  measures  the  tem- 
perature. 

If  M  is  the  mass  at  the  heavier  end  of  the  belt,  m  the  mass 
at  the  lighter  end,  and  F  the  reading  of  the  spring  balance, 
then  the  force  acting  to  oppose  the  rotation  of  the  cylinder  is 
(M  —  m  +  F)  g,  where  g  is  the  acceleration  due  to  gravity. 
The  work  done  in  overcoming  this  force  during  one  revolu- 
tion of  the  cylinder  is  2-rrr  (M  --  m  +  F)  g.  If,  in  n  revolu- 
tions, the  water  of  mass  W  is  raised  from  T1°C.  to  T20C.,  we 
have,  by  equating  the  work  done  and  the  heat  generated. 

2-nrn  (M  —  m  +  F)  g  =  (W  +  w)   (Tz  —  7\  +  R]  J, 

where  w  is  the  water-equivalent  of  the  cylinder  and  the  ther- 
mometer, R  is  a  temperature-correction  to  compensate  for  ra- 
diation, conduction  and  the  viscosity  of  the  water,  and  /  is  the 
mechanical  equivalent  of  heat.  The  purpose  of  the  experi- 
ment is,  by  means  of  this  relation,  to  determine  the  value  of 
J  from  known  and  observed  lvalues  of  the  other  quantities  in- 
volved. 

(a)  Make  a  mixture  of  4  or  5  liters  of     ice-water  whose 
temperature  is  about  10°  below  the  temperature  of  the  room. 
Weigh  out  enough  of  it  to  fill  the  brass  cylinder  about  half 
full.     Suspend  masses  from  the  ends  of  the  belt,  so  that,  when 
the  cylinder  is   rotated  at  moderate   speed,  the  masses  hang 
free  of  the  fixed  parts  of  the  frame. 

(b)  Read  the  temperature  ^  of  the  water,  loosen  the  belt 
so  as  to  eliminate  the  friction  between  it  and  the  cylinder,  and 
perform  100  rotations  of  the  cylinder.     Record  the  final  tem- 
perature t2.     This  is  done  to  determine  the  rate  at  which  the 
temperature  of  the  water  is  changing,  due  to  radiation,  con- 
duction, and  the  viscosity  of  the  water,  just  before  the  test  in 
(c)  is  made. 

(c)  Adjust  the  belt  and  masses  as  they  were  in   (a),  read 


32]  MECHANICAL  EQUIVALENT  OF  HEAT.  71 

the  temperature  7\  of  the  water  and  rotate  the  cylinder  at  the 
same  speed  until  the  temperature  has  risen  about  10°,  and 
again  record  the  temperature.  Record  the  number  of  revolu- 
tions, n. 

(d)  Loosen  the  belt  and  perform  100  rotations,  as  in  (b), 
recording  the  temperatures  £3  and  f4.     This  is  done  to  deter- 
mine the  rate  at  which  the  temperature  is  changing,  due  to 
radiation,  conduction,  and  the  viscosity  of  the  water,  just  after 
the  test  in  (c)  is  made. 

(e)  From  (b)  it  is  evident  that  in  one  revolution  the  fall 
in  temperature  before  the  test,  due  to  influences  other  than  the 
friction  of  the  belt,  is  (^  --  f2)/ioo;  and  from   (d)  the  fall 
per  revolution  after  the  test  is  (t3  —  f4)/ioo.    It  is  reasonable 
to  assume  that,  during  the  test  in   (c)  y  the  average  loss  in 
temperature  per  revolution  is  the  mean  of  the  two.     Hence, 
for  the  «  revolutions  of   (c),  the  loss  in  temperature  due  to 
radiation,  conduction,  and  the  viscosit     of  the  water  is 


From  the  data  and  equations  (i)  and  (2)  determine  the  value 
of/. 

Point  out  the  principal  sources  of  error,  indicating  how  each 
affects  the  result. 

32.     MECHANICAL   EQUIVALENT     OF   HEAT     BY 
PULUJ'S  METHOD. 

This  method  of  determining  the  mechanical  equivalent  of 
heat  involves  the  measurement  of  the  work  done,  through  fric- 
tion, in  raising  a  given  mass  of  mercury  through  a  measured 
range  of  temperature.  The  apparatus  consists  of  two  hollow 
cones,  one  within  the  other,  mounted  on  a  rotating  axle.  Mer- 
cury is  placed  in  the  inner  one  and  a  thermometer  suspended  in 
the  mercury.  The  outer  cone  is  rotated  and  tends,  through  fric- 


72  MECHANICAL  EQUIVALENT  QF  HEAT.  [32 

tion,  to  carry  the  inner  cone  with  it.  This  is  prevented  by  a 
wooden  pointer  attached  to  the  inner  cone.  From  one  end  of 
the  pointer  a  cord  is  stretched  parallel  to  a  line  through  the 
axis  of  the  cones  and  the  axis  of  the  rotator-wheel.  This  cord 
passes  over  a  pulley  to  a  scale-pan  on  which  a  mass  may  be 
placed.  The  deflection  of  the  pointer  may  be  read  by  means  of  a 
scale  under  the  shorter  end  of  the  pointer.  If  the  outer  cone  is 
rotated  with  constant  angular  velocity,  the  pointer  will  be  hela 
at  a  constant  deflection.  It  is  while  the  pointer  is  thus  de- 
flected that  the  stretched  cord  should  be  adjusted  as  stated 
above.  The  force-moment  preventing  the  rotation  of  the  inner 
cone  is  that  due  to  the  tension  in  the  cord,  and  it  may  be 
measured.  The  value  of  this  force-moment  when  multiplied 
by  2?r  gives  a  measure  of  the  work  done  in  each  revolution 
of  the  outer  cone. 

Let  M  be  the  mass  of  the  pan  and  its  contents  and  /  the 
friction  of  the  pulley.  Then  the  total  force  acting  on  the  end 
of  the  pointer  is  (Mg  -f-  /),  where  g  is  the  acceleration  due 
to  gravity.  If  I,  is  the  lever-arm,  the  force-moment  of  this 
force  about  the  axis  of  the  cones  is  L  (Mg  -f-  /).  Now,  during 
each  revolution  of  the  outer  cone  the  work  done  equals  the 
force  of  friction  times  the  circumference  of  the  cone.  But 
it  can  be  easily  shown  that  the  force  of  friction  times  the  cir- 
cumference of  the  cone  is  equal  to  2?r  times  the  force-mo- 
ment of  the  friction.  From  the  fact  that  the  outer  cone  is  in 
equilibrium  under  the  action  of  friction  and  the  tension  in  the 
cord,  it  follows  that  their  force-moments  must  be  equal ;  hence 
the  work  done  in  each  revolution  of  the  outer  cone  is 
2-n-L  (Mg  -f-  f).  If  Wn  is  the  work  done  in  n  revolutions  of 
the  cones,  we  will  have 

J^n..=  27r  ;/  L  (Mg  +  f). 

In  making  «  revolutions  of  the  cones,  let  us  suppose  that  the 
temperature  has  risen  from  7\  to  Tz.  Let  the  sum  of  the 


32]  MECHANICAL  EQUIVALENT  OF  HEAT.  73 

masses  of  the  two  cones  be  mlf  and  the  specific  heat  of  steel 
jx.  Let  m2  be  the  mass  of  the  mercury  in  the  inner  cone,  s2 
its  specific  heat,  and  w  the  water-equivalent  of  the  thermom- 
eter. Then,  if  Q  is  the  quantity  of  heat  in  calories  generated, 

Q  ==  (m^  +  ™2s2  +  w)   [  (T2  --  TO  +  R], 

where  R  is  a  temperature-correction  made  necessary  by  virtue 
of  the  heat  lost  by  the  cones  through  radiation  and  conduction 
to  surrounding  objects.  From  these  values  the  mechanical 
equivalent  of  heat  (/)  may  be  obtained  from  the  relation, 
/  =  W/Q. 

(a)  The  masses  of  the  steel  cones  are  given.     Partially 
fill  the  inner  cone  with  mercury,  but  not  fuller  than  two  centi- 
meters below  the  top,  and  weigh.     Put  the  inner  cone,  ther- 
mometer, pointer,  and  scale-pan  in  position.     Rotate  the  outer 
cone  and  adjust  the  mass  on     the  scale-pan     so  as  to  give 
a  steady  deflection  of  20  degrees  or  more.    When  the  pointer  is 
deflected,  the  pulley  should  be  moved  so  that  the  cord  is  par- 
allel to  the  axis  of  the  machine,  as  already  explained. 

(b)  Read   the   temperature   ^    of   the     mercury,      remove 
the  thermometer  and  pointer   (but  not  the  inner  cone),   and 
perform,  with  the  same  speed  as  before,  enough  rotations  of 
the  wheel  to  make  200  rotations  of  the     cones.     Record  the 
final  temperature,   t2.   This   is   done  to   determine  the  rate  at 
which  the  temperature  is  changing,     due  to     radiation     and 
conduction  before  the  test  in   (c)   is  made. 

(c)  Replace  the  pointer  and  thermometer,  read  the  temper- 
ature, Tj,  and  rotate  the  cone  steadily  as  before  until  the  tem- 
perature has  risen  about  5°,  and  again  record  the  tempera - 
turei  Tz.     Record  the  number  of  rotations,  n,  of  the  cones. 

(d)  Remove  the  pointer  and  thermometer  and  again  per- 
form 200  rotations  of  the  cones,  recording  the  temperatures, 
t?i  and  t4,  as  in  (b).    This  enables  one  to  determine  the  rate  at 


74  COOLING   THROUGH    CHANGE)    OF   STATE.  [33 

which  the  temperature  is  changing,  due  to  radiation  and  con- 
duction, after  the  test  in  (c)  is  made. 

(e)  Detach  the  cord  from  the  end  of  the  pointer  and  attach 
masses  just  sufficient  to  balance  the  mass  of  the  scale-pan 
when  the  cord  is  hung  over  the  pulley.  Now  put  additional 
masses  on  one  side  till  the  system  just  begins  to  move.  Call 
this  extra  mass  p ;  then  /  =  pg,  where  g  is  the  acceleration  due 
to  gravity. 

(/).  From  (b)  the  fall  in  temperature  before  the  test,  due 
to  heat-losses  during  one  rotation,  is  (^  —  f2)/2Oo;  and  from 
(d)  the  fall  per  rotation  after  the  test,  due  to  the  same  cause, 
is  (fa  —  f4)  /2OO.  The  mean  of  these  two  will  fairly  represent 
the  fall  per  rotation  during  the  test  in  (c).  Hence,  for  the  n 
rotations  of  (c),  the  fall  in  temperature  due  to  radiation  and 
conduction  is 


Explain  fully  how  R  is  given  by  the  measurements  of  (b) 
and  (d)  and  the  last  calculation.  Calculate  /  from  the  equa- 
tions previously  given. 

33.     COOLING  THROUGH  CHANGE  OF  STATE. 

Reference. — Duff,    p.    220. 

When  a  solution  changes  from  the  liquid  to  the  solid  state, 
or  vice  versa,  the  temperature  at  which  the  change  occurs  is, 
as  a  rule,  not  as  sharply  defined  in  the  case  of  a  non-crystalline 
substance  as  in  the  case  of  a  crystalline  one.  The  two  types 
of  substances  act  differently,  also,  in  that  supercooling  often 
occurs  in  the  one  case  and  not  in  the  other. 

I.     Melting  Point  and  Cooling  Curve  of  Paraffine. 
(a)      Take  several  pieces  of  capillary  tubing,  each  2  or  3 
cm.  long,  dip  the  ends  in  melted  paraffine,  a  non-crystalline 


33]  COOLING    THROUGH    CHANGE   OF    STATE.  75 

substance,  and  let  them  fill  by  capillarity.  Fasten  them  around 
the  bulb  of  a  thermometer  by  means  of  a  rubber  band.  Place 
the  thermometer-bulb  in  a  small  corked  test-tube,  and  immerse 
and  heat  in  a  water-bath,  taking  care  not  to  allow  any  water 
to  enter  the  test-tube.  Note  the  temperature  at  which  the  par- 
affine  melts.  Remove  the  test-tube  containing  the  thermometer 
and  note  the  temperature  at  "which  the  paraffine  solidifies. 
Take  the  mean  of  these  two  as  the  melting  point. 

(b)  Put  a  thermometer  in  a  test-tube  together  with  enough 
parafi%ie  to  cover  the  bulb  when  melted.  Heat  the  test-tube 
in  a  water-bath  until  the  thermometer  registers  about  70°. 
Remove  the  test-tube  from  the  bath,  clamp  it  in  a  stand,  and 
allow  the  parafffrie  to  cool  slowly  in  air  to  about  38°C.  Record 
the  time  for  each  degree  or  half-degree  fall,  or  at  shorter 
intervals  when  the  cooling  is  evidently  not  uniform.  Plot  on 
coordinate  paper,  using  temperatures  as  ordinates  and  times 
as  abscissae.  Explain  the  form  of  the  different  parts  of  the 
curve,  with  especial  reference  to  rate  of  radiation  and  rela- 
tive specific  heat. 

II.     Cooling  Curve  of  Acetamide. 

Place  a  thermometer  in  a  test-tube  and  surround  the  ther- 
mometer with  crystals  of  acetamide,  filling  the  test-tube  about 
one-third  full.  Place  the  test-tube  in  a  water-bath  and  heat 
to  the  temperature  of  boiling  water.  Remove  the  test-tube 
from  the  bath,  clamp  it  in  a  stand,  and  either  note  the  time 
for  each  degree  or  half-degree  of  fall,  or  record  the  tempera- 
ture for  every  half-minute.  Continue  the  readings  until  the 
temperature  falls  to  about  4O°C.  Plot  on  coordinate  paper, 
using  temperatures  as  ordinates  and  times  as  abscissae.  Ex- 
plain the  form  of  the  different  parts  of  the  curve.  Compare 
with  the  cooling  curve  of  paraffine. 


76  HEAT    OF    FUSION.  [34 

34.     HEAT  OF  FUSION. 

The  purpose  of  this  experiment  is  to  determine  the  heat  of 
fusion  of  the  alloy  knozvn  as  Wood's  fusible  alloy. .  Its  com- 
position is  lead  25.9  parts,  cadmium  7  parts,  bismuth  52.4 
parts,  and  tin  14.7  parts.  The  alloy  is  a  solid  at  ordinary  tem- 
peratures, but  readily  melts  in  hot  water.  The  method  em- 
ployed will  be  the  method  of  mixtures.  A  known  mass  of  the 
metal  is  placed  in  a  nickle  crucible  of  known  mass  and  specific 
heat,  suspended  in  a  copper  cage  of  known  mass  and  spe- 
cific heat,  and  is  heated  to  the  temperature  of  boiling  water. 
The  whole  is  then  plunged  into  a  known  quantity  of  cold 
water  in  a  calorimeter  cup  and  the  change  in  temperature 
noted.  The  following  changes  occur  wherein  heat  is  given 
out:  (i)  The  alloy  cools  as  a  liquid  from  the  temperature  of 
the  hot  water-bath  down  to  the  freezing  point  of  the  alloy,  (2) 
the  alloy  changes  from  a  liquid  to  a  solid  without  change  of 
temperature,  (3)  the  alloy  cools  as  a  solid  from  its  freezing 
point  down  to  the  final  temperature  of  the  mixture  in  the 
calorimeter,  (4)  meanwhile  the  nickle  crucible  and  the  cop- 
per cage  cool  from  the  temperature  of  the  hot  water-bath  to 
the  final  temperature  of  the  mixture.  The  changes  wherein 
heat  is  absorbed  are  those  accompanying  the  rise  in  tempera- 
ture of  the  calorimeter  cup  and  contents  from  the  initial  tem- 
perature of  the  cold  water  up  to  the  final  temperature  of  the 
mixture.  From  these  data,  if  the  specific  heat  of  the  metal 
in  the  solid  and  in  the  liquid  state  be  known,  the  heat  of  fusion 
may  be  found.  Let  M  be  the  mass  of  the  alloy;  ;;/,  the  mass 
of  the  nickel  crucible  ;  IV,  the  mass  of  the  water  in  the  calor- 
imeter cup  plus  the  water-equivalent  of  the  cup,  thermometer, 
and  stirrer;  slf  the  specific  heat  of  the  liquid  alloy;  s2,  that  of 
the  solid  alloy;  s3,  that  of  the  nickel  crucible;  T,  the  melting 
point  of  the  alloy;  tlf  the  initial  temperature  of  the  alloy  and 
crucible ;  £0,  the  initial  temperature  of  the  calorimeter  and  con- 


34]  HEAT   OF   FUSION.  77 

tents;  t,  the  final  tempeiature;  and  L,  the  heat  of  fusion  per 
gram  of  the  alloy.  Write  the  proper  equation  representing 
the  transfer  of  heat  in  the  above  process,  using  the  symbols 
indicated,  and  solve  the  equation  for  L. 

(a)  First  determine  the  mass,  in  grams,  of  those  things 
whose  mass  it  is  necessary  to  know.  Place  the  alloy  in  the 
crucible  and  determine,  wifhin  3°,  its  melting  point.  To  do 
this,  stand  the  crucible  (in  a  clamp  provided  for  it)  in  a  ves- 
sel of  water.  Heat  the  water,  taking  care  after  the  water  has 
reached  a  temperature  of  60°  that  the  heating  be  done  slowly, 
so  that  the  alloy  will  be  at  the  same  temperature  as  the  water. 
The  thermometer  must  be  placed  in  the  water  and  not  in  the 
alloy.  The  liquid  alloy  "wets"  glass,  henc >  some  of  it 
would  be  withdrawn  with  the  thermometer  and  relatively 
large  changes  in  mass  introduced.  If  the  thermometer  were 
placed  in  the  alloy  and  left  there,  there  would  be  great  danger 
of  its  breaking  on  the  solidification  of  the  metal.  After  the 
melting  point  of  the  alloy  has  been  found,  bring  the  water  to 
the  boiling  point,  taking  care  that  no  water  gets  inside  the 
crucible.  Remove  the  crucible,  noting  the  time,  and  quickly 
wipe  the  outside ;  then  quickly  an  I  carefully  lower  it  into  the 
calorimeter,  right  side  up,  with  its  contained  alloy,  letting 
the  water  run  into  the  crucible  and  thus  more  quickly  cool  the 
alloy.  Only  a  few  seconds  should  elapse  between  the  re- 
moval of  the  alloy  from  the  hot  water-bath  and  its  immer- 
sion in  the  calorimeter.  Stir  the  mixture  continuously,  not- 
ing the  time  and  temperature  when  the  mixture  becomes 
uniform  and  starts  to  cool,  and  then  again  5,  10  and  15  min- 
utes later. 

By  plotting  times  as  abscissae  anij  temperatures  as  ordin- 
ates,  find  the  temperature  which  the  mixture  should  have  if 
its  temperature  could  be  made  uniform  the  instant  the  alloy 
enters  it.  The  effect  of  radiation  is  thus  accounted  for.  A 
little  consideration  will  make  it  evident  that  the  cooling  curve 


78  HEAT    OF   VAPORIZATION    AT    BOILING    POINT.  [3$ 

in  the  plot,  prolonged,  backward  to  intersection  with  the  axis 
of  ordinates  will  give  the  correct  temperature. 

The  specific  heats  of  the  solid  and  liquid  alloy  will  be  given 
in  the  Tables,  or  if  time  permits  they  may  be  found  by  the 
method  of  mixtures.  The  specific  heat  of  nickel  and  copper 
may  be  found  in  the  Tables. 

Make  two  or  three  determinations,  as  outlined  above,  of  the 
heat  of  fusion  of  Wood's  alloy. 

(b)  Why  is  only  a  rough  determination  of  the  melting  point 
of  the  alloy  necessary?  Discuss  the  relative  accuracy  with 
which  the  different  masses  used  must  be  determined  in  order 
that  the  precision  of  measurement  of  the  result  may  be  2  per 
cent.  Point  out  the  principal  sources  of  error  in  the  experi- 
ment. 


35.   HEAT  OF  VAPORIZATION  AT  BOILING  POINT. 
References. — Edser,  p.  152;  Watson's  Practical  Physics,  p.  237. 

In  this  experiment  Kahlenberg's  modification  of  Berthelot's 
apparatus1  is  used. 

(a)  Determine  the  boiling  point  of  the  liquid  used,  by  care- 
fully  heating  a   small   quantity   in   a  test-tube  or  beaker   by 
means  of  a  water-bath. 

(b)  Weigh  the  calorimeter,  first  dry  and  empty,  then  about 
two-thirds  full  of  water.     Carefully  dry  and  weigh  the  worm, 
together  with  the  two  corks  which  fit  its  ends.     Set  up  the 
calorimeter  with  stirrer,  worm,  and  thermometer.     The  boiler 
consists  of  a  test-tube  to  which  is  fitted  a  rubber  stopper.    A 
glass  tube  extends  through  the  stopper  to  the  bottom  of  the 
test-tube;  two  wires  also  pass  through  the  stopper,  and  are 
connected  to  a  coil  of  wire  which  loosely  surrounds  a   part 
of  the   glass   tube.     When   in   use   the  test-tube   is   inverted, 

journal  of  Physical  Chemistry,  1901,  Vol.  5,  p.  215. 


35]  HEAT   OF   VAPORIZATION    AT   BOILING   POINT.  79 

enough  liquid  being  placed  in  it  to  completely  cover  the  coil 
of  wire  after  the  tube  is  inverted.  An  electric  current  is  then 
sent  through  the  coil,  furnishing  the  heat  to  boil  the  liquid. 
The  vapor  from  the  boiling  liquid  passes  downward  through 
the  glass  tube  and  enters  the  worm,  when  the  boiler  is  placed 
in  position  over  the  calorimeter. 

Care  should  be  taken  to  use  enough  liquid  so  that  the  heat- 
ing coil  is  covered  throughout  the  experiment.  Never  allow 
the  heating  current  to  be  closed  through  the  coil  while  the 
coil  is  not  completely  covered  with  liquid.  Do  not  place  the 
boiler  over  the  calorimeter  until  the  liquid  boils  and  the  vapor 
is  issuing  freely  from  the  tube.  See  that  the  cork  is  removed 
from  the  free  end  of  the  worm,  as  the  boiling  must  be  done 
at  atmospheric  pressure,  otherwise  the  temperature  of  the 
vapor  will  not  be  that  found  in  (a). 

When  all  is  ready,  note  the  temperature  of  the  calorimeter, 
and  place  the  boiler  in  its  proper  place  so  that  the  vapor  enters 
the  worm.  Gently  stir  the  water  in  the  calorimeter,  and  read 
the  thermometer  at  one-minute  intervals  until  the  temperature 
has  risen  about  5°.  Turn  off  the  current,  remove  the  boiler, 
cork  the  ends  of  the  worm,  and  continue  to  read  the  ther- 
mometer at  one-minute  intervals  for  five  minutes.  Remove 
the  worm  from  the  calorimeter,  carefully  dry  the  outside,  and 
weigh.  Pour  the  contents  of  worm  and  boiler  into  the  proper 
bottle,  and  empty  the  calorimeter.  See  that  the  electric  cir- 
cuit is  disconnected. 

(c)  From  the  series  of  temperatures  taken  determine  the 
rise  of  temperature  of  the  calorimeter,  correction  being  made 
for  radiation.  Determine  the  wacer-equivalent  of  the  calor- 
imeter and  contents,  including  the  stirrer,  thermometer,  empty 
worm,  and  water.  The  necessary  specific  heats  may  be  ob- 
tained from  the  Tables.  Calculate  the  amount  of  heat  gained 
by  the  calorimeter.  Knowing  the  mass  of  the  vapor  con- 
densed, the  change  in  temperature  of  the  liquid,  and  the  spe- 


80  HEAT   OF   VAPORIZATION   AT    ROOM    TEMPERATURE.  [ 36 

cific  heat  of  the  liquid  (see  the  Tables  for  the  specific  heat), 
calculate  the  heat  transferred  to  the  calorimeter,  and  deter- 
mine the  heat  of  vaporization  of  the  liquid  at  its  boiling  point 

36.  HEAT  OF  VAPORIZATION  AT  ROOM  TEM- 
PERATURE. 

Reference. — Duff,    p.    233. 

The  heat  of  vaporization  of  a  liquid  varies  with  the  temper- 
ature at  which  vaporization  takes  place.  In  nature,  vaporiza- 
tion takes  place,  for  the  most  part,  at  atmospheric  tempera- 
ture rather  than  at  boiling  temperature.  The  object  of  this 
experiment  is  to  find  the  amount  of  heat  necessary  to  vapor- 
ize  one  gram  of  a  liquid  at  the  room  temperature.  To  do 
this,  dry  air  is  made  to  bubble  through  the  liquid,  thus  in- 
creasing the  free  surface  and  producing  rapid  evaporation. 
The  loss  of  weight  of  the  liquid  gives  the  amount  evaporated, 
while  from  the  fall  of  temperature  of  the  liquid  and  calori- 
meter, together  with  their  masses  and  specific  heats,  the  heat- 
loss  can  be  determined  and  the  heat  of  vaporization  calcu- 
lated. 

(a)  Carefully  weigh  the  calorimeter  cup  when  dry  and 
empty,  and  again  when  containing  about  100  grams  of  alcohol. 
Place  the  cover  on  the  calorimeter,  with  the  thermometer- 
bulb  in  the  liquid  and  arranged  so  that  dry  air  can  be  forced 
through  the  liquid  by  means  of  a  small  foot-bellows.  Have 
the  initial  temperature  of  the  liquid  2°  or  3°  above  the  room 
temperature.  Pass  the  dry  air  gently  through  the  liquid,  al- 
lowing ample  room  for  the  vapor-charged  air  to  escape,  un- 
til the  temperature  is  as  much  below  room  temperature  as 
the  initial  temperature  was  above  it.  If  the  air  is  forced  too 
rapidly  through  the  liquid,  not  all  of  the  liquid  which  is  car- 
ried away  will  be  vaporized.  Weigh  the  calorimeter  and  re- 
maining liquid.  A  50°  thermometer  graduated  in  tenths  of 


37]  FREEZING   POINT  OF   SOLUTIONS.  Ri 

a  degree  should  be  used.  Wet  the  thermometer,  with  the  li- 
quid used,  about  as  high  as  the  depth  to  which  it  will  be 
placed  in  the  liquid  in  the  calorimeter,  so  that  as  much  liquid 
will  be  introduced  at  first  as  will  be  withdrawn  later  when  the 
thermometer  is  removed  from  the  calorimeter. 

(b)  Repeat    (a)   two  or  three     times.       When     finished, 
empty  and  dry  the  calorimeter.   If  a  liquid  other  than  water 
was  used,  it  should  be  poured  back  into  its  proper  bottle. 

(c)  From  the   amount   of   liquid   evaporated,   the   fall   in 
temperature,   and   the   water-equivalent   of   the    thermometer, 
calorimeter,  and  liquid  used,  determine  the  heat  of  vaporiza- 
tion in   (a)   and   (fr),  taking  the  mean  as  the  final  value.     It 
will  be  necessary  to  assume  that  the  heat,  used  up  in  vapor- 
izing the  liquid,  all  came  from  the  calorimeter  and  its  contents. 
The  mean  of  the  initial  and  final  amounts  of  liquid  in  the  calor- 
imeter should  be  taken  as  the  amount  of  liquid  cooled. 

(d)  Point  out  the  chief  sources  of  error. 

Give  a  reason  why  the  value  of  the  heat  of  vaporization  of 
a  liquid  increases,  when  the  temperature,  at  which  the  vapori- 
zation of  that  "liquid  occurs,  is  lowered. 

Evaporation  takes  place  from  dry  ice  at  temperatures  be- 
low the  freezing  point.  This  change  from  solid  directly  to 
vapor  is  called  sublimation.  By  what  amount  would  you  ex- 
pect the  heat  of  sublimation  of  ice  at  o°C.  to  exceed  the  heat 
of  vaporization  of  water  at  o°C.? 


37.     FREEZING  POINT  OF  SOLUTIONS. 

References. — Watson,   p.   268;    Watson's   Practical    Physics,   p.   258; 

Edser,  p.  167. 

The  object  of  this  experiment  is  (i)  to  observe  the  lowering 
of  the  freezing  point  of  water  caused  by  dissolving  table  salt 
and  sugar  in  it  to  form  solutions  of  different  concentrations. 


82  FREEZING   POINT  OF   SOLUTIONS.  [37 

and  (2)   to  determine  the  molecular  weights  of  the  salt  and 
the  sugar  by  means  of  this  lowering. 

(a)  Using  a  50°   thermometer,   determine     the     freezing 
point  of  pure  water  with  the  same  apparatus  as  that  employed 
in  the  calibration  of  the  100°  thermometer.  Then  determine  the 
freezing  point  of  a  4  per  cent  solution  of  common  salt  in  water. 
By  percentage  solution  is  here  meant  the  number  of  grams  of 
dissolved  substance  per   100  grams  of  the  solution.     Repeat 
for  an  8  per  cent  and  for  a  12  per    cent     solution.     Choose 
such  amounts  of  these  solutions  in  the  three  cases  as  will  con- 
tain the  same  mass  of  the  solvent,  for  example,  100  gm.  of 
the  4  per  cent,  104.3  Sm-  °f  the  8  Per  cent>  and  109.1  gm.  of 
the  12  per  cent  solution. 

(b)  Repeat  (a)  with  aqueous  solutions  of  sugar  of  6,  12. 
and  1 8  per  cent  concentration. 

(c)  Tabulate  the  results  of  (a)  and  of  (b),  and  for  each 
of  the  six  cases  calculate  the  lowering,  per  gram  of  dissolved 
substance,  of  the  freezing  point  of  a    .given  mass     oi     water. 
What  relation  seems  to  hold  between  the  change  of  freezing 
point  of  a  given  mass  of  water  and  the  mass  of  dissolved  sub- 
stance ? 

Note  the  difference  of  freezing  points  for  12  per  cent  solu- 
tions of  table  salt  and  sugar. 

(d)  Calculate  the  molecular  weights  (M)  of  table  salt  and 
sugar  from  the  relation  M  =  Ks/St,  where  s  is  the  number  of 
grams  of  dissolved  substance,  5  is  the  number  of  grams  of  the 
solvent,  t  is  the  depression  of  the  freezing  point,  and  K  is  a 
constant,  whose  value  is  1850  for  aqueous  solutions. 

Water  in  dilute  solutions  is  thought  to  have  the  power  of 
breaking  up  the  molecules  of  some  dissolved  substances  into 
ions,  each  ion  having  the  same  effect  in  lowering  the  freezing 
point  as  a  molecule  has.  The  result  of  this  is  that  the  observed 
lowering  of  the  freezing  point  of  water  is  three  times  as  great 
in  the  case  of  some  dissolved  substances  and  twice  as  great  in 


38]  HEAT    OF    SOLUTION.  83 

the  case  of  others  as  the  value  calculated  by  the  formula. 
What  ionizing  effect,  if  any,  has  water  on  the  table  salt  and 
sugar  ? 

What  relative  lowering  of  freezing  point  of  equal  masses 
of  water  would  you  expect  (a)  if  equal  numbers  of  molecules 
of  table  salt  and  sugar  were  brought  into  solution,  (&)  if  equal 
masses  ? 


38.     HEAT  OF  SOLUTION. 

The  quantity  of  heat  absorbed  in  the  solution  of  one  gram 
of  a  substance  in  a  large  amount  of  the  solvent  is  called  its 
heat  of  solution.  If  heat  is  given  out  in  the  solution,  the 
quantity  is  considered  negative. 

If  the  temperature  of  the  salt  after  solution  be  different 
from  that  at  which  it  was  poured  into  the  water,  it  will  be 
necessary  to  consider  its  specific  heat  also.  According  to  the 
following  method  the  heat  of  solution  and  the  specific  heat 
are  both  determined,  although  the  former  is  the  main  object 
of  the  experiment. 

(a)  On  one  of  the  Becker  balances  weigh  out  on  pieces 
of  dry  paper  two  portions  of  salt,  each  of  about  15  grams,  to 
o.oi   gram.     Make  sure  that  the  salt  is  quite  dry  and  finely 
pulverized,  and  be  careful  not  to  leave  any  in  the  balance-pan. 
This  amount  of  salt,  if  sodium  hyposulphite  be  used,  when 
dissolved  in  200  grams  of  water  will  lower  its  temperature 
a  little  over  3°.     It  is  best  to  have  the  cup  about  3°  warmer 
than  the  jacket,  because  the  larger  part  of  the  salt  dissolves 
in  a  few  seconds,  so  that  the  loss  of  heat  by  radiation  during 
this  time  is  small ;  and  the  temperature  being  then  reduced 
to  about  that  of  the  jacket,  there  is  no  loss  by  radiation  dur- 
ing the  longer  time  required  for  the  complete  solution  of  the 
salt. 

(b)  Set  up  the  calorimeter,   with  the  jacket  filled   with 


84  HEAT   OF   SOLUTION.  [38 

water  at  the  room  temperature,  and  the  cup  containing  200 
grams  of  water  about  3°  warmer.  Keep  the  stirrer  moving 
slowly  and  read  the  temperature  of  the  cup  at  intervals  of 
one  minute  for  about  five  minutes.  Pour  in  the  salt  one 
minute  after  the  last  observation,  stir  rather  vigorously  to 
hasten  solution,  and  record  the  final  temperature. 

From  the  series  of  observations,  calculate  the  temperature 
of  the  cup  at  the  time  when  the  salt  was  poured  in.  The  tem- 
perature of  the  salt  at  that  time  may  be  assumed  to  be  that  of 
the  room. 

(c)  Make  a  similar  trial  with  a  second  portion  of  salt, 
having  the  cup  at  about  4O°C.     Make  sure  that  there  is  the 
proper  difference  between  cup  and  jacket  at  the  time  the  salt 
is  poured  in.     To  determine  the  water-equivalent  of  the  calor- 
imeter cup  and  stirrer,  it  will  be  necessary  to  know  their  masses 
and  the  specific  heats  of  the  metals  of  which  they  are  made. 
The  water-equivalent  of  the  thermometer  may  be  calculated 
from  the  number  of  cc.  which  it  displaces  when  immersed  in 
a  graduate  to  the  proper  depth. 

(d)  Call  the  specific  heat  of  the  salt  x,  and  its  heat  of  so- 
lution in  water  y.     Write  for  each  of  the  cases  (b)  and  (c) 
an  equation  involving  the  following  quantities : 

1.  Heat  lost  by  water  in  cup. 

2.  Heat  lost  by  cup,  stirrer,  and  thermometer. 

3.  Heat  gained  or  lost  by  salt  in  changing  temperature. 

4.  Heat  absorbed  during  solution  of  salt.    It  will  be  well  to 
assume  that  the  salt  initially  is  at  the  room  temperature  in  the 
two  cases. 

Solve  the  two  equations  for  .1-  and  y. 

Why  should  the  value  of  the  heat  of  solution  as  obtained 
by  this  method  be  proportionately  more  accurate  than  that 
for  the  specific  heat  ? 

Caution : — Do  not  leave  the  solution  standing  in  the  cup. 
Wash  it  out  as  soon  as  possible. 


39]  HEAT    OF    NEUTRALIZATION.  85 


39.     HEAT  OF  NEUTRALIZATION. 

When  an  aqueous  solution  of  a  strong  acid  is  poured  into 
an  aqueous  solution  of  a  strong  alkali  until  a  neutral  mixture 
is  formed,  the  essential  chemical  reaction  which  occurs  is  the 
formation  of  water.  For  instance,  if  aqueous  solutions  of  hy- 
drochloric acid  and  sodium  hydroxide  are  made  to  form  a 
neutral  mixture,  although  the  mixture  is  a  solution  of  sodium 
chloride  (table  salt),  the  only  chemical  reaction  occurring  is 
the  formation  of  water.  The  heat  generated  is  called  the  heat 
of  neutralization.  The  object  of  the  present  experiment  is 
to  determine  the  heat  of  neutralization  corresponding  to  the 
formation  of  a  gram-molecular  weight  of  water.  In  the  case 
just  mentioned,  this  will  occur  when  1000  gm.  each  of  normal 
solutions  of  the  acid  and  the  alkali  are  mixed. 

A  0.5  normal  solution  of  each  of  the  above  compounds  is 
furnished.  By  a  normal  solution  is  meant  one  which,  in  1000 
cubic  centimeters  of  the  solution,  contains  a  mass  of  the  com- 
pound (which  is  to  enter  into  the  new  combination)  equal  in 
grams  to  its  molecular  weight.  Thus  the  normal  solution  of 
sodium  hydroxide  is  a  solution  which  contains,  in  1000  cc.  of 
the  solution,  40  gm.  (23+16+1)  of  sodium  hydroxide, 
or  23  gm.  of  sodium,  16  gm.  of  oxygen,  and  I  gm.  of  hydro- 
gen. The  0.5  normal  solution  contains  one-half  as  much  in 
the  same  volume  of  solution. 

It  is  evident  that  if  equal  volumes  of  these  solutions  be 
mixed,  the  reaction  will  be  just  completed,  and  the  result  will 
be  a  neutral  solution  of  sodium  chloride.  The  solutions  are 
to  be  mixed  in  the  calorimeter  cup  at  as  nearly  as  possible 
the  same  temperature,  and  the  resulting  rise  of  temperature 
noted.  The  alkali  should  be  placed  in  the  cup,  and  the  acid 
added  to  it.  The  acid,  being  immediately  neutralized,  will 
then  have  no  action  on  the  metal  of  the  cup. 


86  COEFFICIENT  OF  EXPANSION  OF  AIR.  [40 

(a)  Measure  out  100  cc.  of  the  sodium  hydroxide  solution 
in  the  cup,  and  the  same  volume  of  the  hydrochloric  acid  solu- 
tion in  the  beaker.    Wet  the  inside  of  the  beaker  with  the  acid 
solution  before  pouring  the  measured  amount  into  it.    This  is 
to  compensate  for  the  liquid  which  remains     in     the     beaker 
when  later  it  is  emptied.    A  small  error  is  introduced  by  taking 
the  second  thermometer  out  of  the  beaker  after  reading  its 
temperature,  but  this  may  be  neglected. 

If  care  has  been  taken  not  to  handle  the  cup  and  beaker 
any  more  than  is  necessary,  the  two  temperatures  should  be 
very  nearly  the  same  when  ready  for  use.  It  may  safely  be 
assumed  that  the  resulting  solution  of  sodium  chloride  has 
risen  to  the  final  temperature  from  the  mean  of  the  two  initial 
temperatures. 

A  direct  determination  of  the  specific  heat  of  the  sodium 
chloride  solution  is  impracticable.  The  value,  0.987,  which 
has  been  calculated  by  interpolation  from  tabulated  results, 
may  be  used  for  this  case. 

Make  two  trials,  and  calculate  for  each  the  quantity  of 
heat  which  would  have  been  evolved  if  1000  cc.  of  normal  so- 
lution had  been  used  in  each  case.  Will  this  cause  the  forma- 
tion of  one  gram-molecular  weight  of  water? 

(b)  Repeat  the  work,  if  there  is  time,  with  solutions  of 
potassium  hydroxide  and  sulphuric  acid,  and  compare  the  re- 
sults with  that  in  (a). 


40.     COEFFICIENT  OF  EXPANSION     OF     AIR     AT 
CONSTANT  PRESSURE  BY  FLASK  METHOD. 

The  purpose  of  this  experiment  is  to  determine  the  coefficient 
of  expansion  of  air  by  observing  the  contraction  (inside  of  a 
glass  flask  or  bulb)  of  a  given  mass  of  air  when  its  tempera- 
ture is  lowered  a  measured  amount. 


40]  COEFFICIENT  OF  EXPANSION  OF  AIR.  87 

A  glass  bulb  with  a  long  tubular  neck  closed  by  a  stop-cock 
is  suspended  in  a  steam  bath  to  bring  the  enclosed  air  to  the 
temperature  of  boiling  water.  The  stop-cock  is  then  closed, 
imprisoning  in  the  bulb  a  given  mass  of  air  at  atmospheric 
pressure.  The  bulb  is  then  inverted  and  plunged  into  a  bath 
of  ice-water,  the  stop-cock  is  opened,  and  the  enclosed  air  is 
brought  again  to  atmospheric  pressure.  The  apparent  volume 
of  the  given  mass  of  air  when  contracted  is  found  by  determin- 
ing the  difference  between  the  masses  of  ice-water  filling  the 
whole  bulb  and  that  part  of  the  bulb  not  occupied  by  the  con- 
tracted air. 

Let  V2  and  V±  represent  the  volumes  of  the  enclosed  air  at 
the  temperature  t2  of  the  steam  and  the  temperature  /t  of 
the  ice-water,  respectively.  Let  V2  be  the  volume  of  the 
whole  bulb  as  determined  by  filling  with  ice-water.  If  a  is  the 
coefficient  of  expansion  of  air  at  constant  pressure  and  y  the 
coefficient  of  cubical  expansion  of  glass,  we  will  have,  by  the 
definitions  of  a  and  y, 

(1)  V,=    V,    [I    +a    (*,--.*.)    ], 

(2)  V,=-   V't   [I    +  y   (V--*,)    ]. 
Solving  for  a,  we  get 

*V-  v,     _^r 
~^iU-'i)      vj' 

If  the  temperature  tl  of  the  ice- water  is  not  o°C.,  the  value  of 
a  obtained  from  (3)  will  not  be  referred  to  standard  initial 
temperature  and  the  result  cannot,  therefore,  be  compared  with 
the  value  given  in  the  Tables. 

(a)  Thoroughly  dry  the  bulb  by  rinsing  out  ten  times  or 
more  with  dry  air.  This  is  done  by  alternately  exhausting  by 
means  of  the  jet-pump  and  admitting  dry  air  from  the  room. 
Then  hang  it  in  the  boiler  with  the  bulb  down  and  with  the 


88  COEFFICIENT  OF  EXPANSION  OF  AIR.  [40 

stop-cock  open.  If  there  is  any  chance  for  the  steam  to  enter, 
attach  a  rubber  tube  to  the  open  end  and  place  the  other  end 
of  this  tube  where  the  steam  can  not  enter.  Boil  the  water, 
causing  the  steam  to  pass  around  the  bulb  until  the  air  inside 
it  is  at  the  temperature  of  the  steam.  Then  introduce  the  stop- 
cock thinly  coated  with  grease,  close  the  cock,  remove  from  the 
boiler,  and  allow  to  cool.  (Care  is  necessary  here  to  keep  the 
bulb  air-tight  and  at  the  same  time  to  keep  the  stop-cock  from 
breaking  when  it  is  cooled.)  Next  place  it  under  the  surface 
of  the  ice-water,  and  open  the  stop-cock,  under  water,  allowing 
the  water  to  enter  but  not  the  air  to  escape.  After  allowing 
time  enough  for  the  bulb  to  come  to  the  temperature  of  the 
ice-water,  raise  the  bulb  so  that  the  level  of  the  water  inside 
is  the  same  as  that  without,  thus  assuring  the  same  pressure 
in  the  enclosed  air  as  before.  Close  the  stop-cock,  remove  and 
dry,  and  then  carefully  weigh.  In  order  to  obtain  the  volume, 
the  bulb  must  now  be  weighed  full  of  water,  and  then  again 
empty  and  dry.  It  is  best  to  fill  with  ice-water  and  to  make 
the  weighings  when  it  is  cold,  so  as  to  get  the  volume  at  o°C. 
In  drying  the  bulb  great  care  should  be  taken  not  to  break 
the  stop-cock  by  the  heat.  These  weighings  will  enable  you 
to  determine  V2,  the  inside  volume  of  the  whole  bulb  at  o°C., 
and  V r,  the  volume  of  the  air  in  the  bulb  when  under  atmos- 
pheric pressure  and  at  o°C. 

From  the  results  of  the  above  measurements  and  the 
coefficient  of  expansion  of  glass  (see  the  Tables)  find  the 
coefficient  of  expansion  of  air. 

(b)  If  time  permits,  repeat  with  some  available  gas  other 
than  air,  and  compare  the  result  with  that  of  air. 


41  ]  CONSTANT-PRESSURE  AIR-THERMOMETER.  89 

41.     EXPANSION    OF  AIR.     CONSTANT-PRESSURE 
AIR-THERMOMETER. 

References. — Edser,   p.    108;    Duff,   p.    187. 

The  object  of  this  experiment  is  (i)  to  determine  the 
mean  coefficient  of  expansion,  between  o°C.  and  ioo°C.,  of  Air 
at  constant  pressure,  by  means  of  the  constant-pressure  air- 
thermometer;  and  (2)  to  test  the  gas-lazvs.  In  the 
form  of  air-thermometer  used  dry  air  is  contained  in 
a  glass  tube  graduated  in  cc.  and  closed  at  one 
end.  The  graduated  tube  is  connected  to  an  open 
glass  tube  by  rubber  tubing,  the  whole  forming  a  "U" 
containing  mercury.  The  pressure  on  the  enclosed  air  can  be 
regulated  to  any  desired  constant  value  by  raising  or  lowering 
the  open  glass  tube.  Surrounding  the  graduated  tube  con- 
taining the  air  is  a  vessel,  covered  by  an  asbestos  jacket,  in 
which  a  water-bath  may  be  placed  or  through  which  steam  may 
be  passed.  The  graduated  tube  is  vertically  adjustable  through 
a  sleeve  in  the  bottom  of  the  vessel,  so  that  the  meniscus  of  the 
mercury  may  be  seen  outside  and  the  volume  read. 
Coefficient  of  Expansion. 

(a)  Fill  the  vessel  with  a  mixture  of  ice  and  water,  and, 
when  the  enclosed  air  has  had  time  to  come  to  the  tempera- 
ture of  the  bath,  adjust  the  pressure  so  that  it  is  10  cm.  less 
than  atmospheric  pressure,  and  read  the  volume. 

Fill  the  vessel  with  water  at  io°C.,  adjust  the  pressure  to 
the  same  value  as  before,  and  again  read  the  volume.  In  this 
way  raise  the  temperature  by  steps,  reading  the  volume  of  the 
air  at  10°,  20°,  30°,  45°,  60°,  80°,  taking  care  each  time  to 
wait  long  enough  (three  minutes  or  more)  for  the  enclosed 
air  to  come  to  the  same  temperature  as  the  bath,  and  each 
time  adjusting  the  pressure  so  that  it  is  10  cm.  less  than  at- 
mospheric pressure.  The  mercury  meniscus  on  the  closed- 
tube  side  should  always  be  as  close  to  the  bottom  of  the  jacket 


90  CONSTANT-PRESSURE  AIR-THERMOMETER.  [4! 

as  will  just  permit  of  reading  the  meniscus.     This  is  done  so 
that  the  enclosed  air  will  not  be  outside  the  water-bath. 

Empty  the  vessel,  place  a  cover  over  the  top,  and  pass 
steam  through  the  vessel,  in  at  the  bottom  and  out  at  the  top. 
Use  two  Bunsen  burners,  if  necessary,  to  obtain  an  abundant 
flow  of  steam.  After  waiting  five  or  ten  minutes  for  the  enclosed 
air  to  reach  the  temperature  of  the  steam,  take  another  read- 
ing of  the  volume,  the  pressure  conditions  being  the  same  as 
before. 

(b)  Make  another  and  similar  series  of  observations  at  a 
pressure  10  or  20  cm.  above  atmospheric  pressure. 

(c)  Plot  the  observations  of  (a)  and  of   (b)  on  the  same 
sheet   of   coordinate   paper,   using  temperatures    (centigrade) 
as  abscissae  and  volumes  as  ordinates. 

From  the  volume  at  o°C.  and  the  volume  at  TOO°C.,  as 
taken  from  the  curve,  calculate,  for  each  curve,  the  average 
apparent  coefficient  of  expansion  of  the  air  between  those 
temperatures.  Take  the  mean  of  the  two  results,  correct  for 
the  expansion  of  the  glass,  and  obtain  /?,  the  absolute  coeffi- 
cient of  expansion  of  air. 

Gas  Laws  Tested. 

(d)  From  the  two  curves  in  (e)  find  out  by  proportion  if  the 
volume  of  a  gas  varies  directly  as  the    absolute    temperature 
when  the  mass  and  pressure  of  the  gas  remain  constant. 

By  taking  some  particular  volume  and  noting  the  tempera- 
ture and  pressure  -on  each  curve  corresponding  to  that  volume 
find  out  by  proportion  if  the  pressure  of  a  gas  varies  directly  as 
the  absolute  temperature  when  the  mass  and  volume  of  the  gas 
remain  constant. 

Similarly,  by  taking  some  particular  temperature  on  the  two 
curves  and  noting  the  volume  and  pressure  on  each  curve  cor- 
responding to  that  temperature,  find  out  if  the  volume  of  a  gas 
varies  inversely  as  its  pressure  when  the  mass  and  tempera- 
ture of  the  gas  remain  constant. 


42]  CONSTANT-VOLUME  AIR-THERMOMETER.  C)t 

42.     CONSTANT-VOLUME   AIR-THERMOMETER. 
References. — Duff,  p.  186;  Millikan,  p.  125. 

The  object  of  this  experiment  is  to  study  the  law  of  varia- 
tion of  the  pressure  of  a  given  mass  of  enclosed  air  whose 
volume  is  kept  constant  while  its  temperature  is  changed. 
The  air  is  enclosed  in  a  glass  bulb  mounted  on  the  frame  used 
in  Exps.  4  and  5.  The  frame  is  placed  near  a  table  so  that 
the  bulb  may  be  surrounded  by  a  water-bath,  by  shaved  ice, 
or  by  a  steam-bath,  the  table  and  an  iron  stand  being  made 
use  of  to  support  each  bath  in  turn.  A  thermometer  is 
placed  in  the  bath  to  give  its  temperature.  The  pressure  on 
the  enclosed  gas  is  regulated  by  raising  or  lowering  the  open 
tube,  as  is  done  in  the  experiment  on  Boyle's  law.  The  value 
of  this  pressure  may  be  determined  from  the  barometer-read- 
ing and  the  difference  in  the  levels  of  the  mercury  on  the  two 
sides  of  the  frame.  Each  time  before  taking  the  readings, 
the  volume  of  the  air  in  the  bulb  is  made  the  same  by  bring- 
ing the  mercury  meniscus  to  the  level  of  the  wire  point  inside 
the  glass  tube  attached  to  the  bulb. 

Caution : —  The  mercury  on  the  bulb  side  should  always  be 
lowered  some  distance  before  changing  to  a  lower  temperature. 
Be  especially  careful  to  do  this  before  removing  the  steam- 
bath  when  you  have  taken  a  reading  at  the  boiling  point; 
otherwise,  on  cooling,  the  mercury  will  run  into  the  bulb.  Do 
not  hurry  in  taking  the  readings  after  changing  the  tempera- 
ture, but  wait  until  the  meniscus  set  at  the  wire-point  remains 
stationary. 

(a)  Without  any  bath  in  the  reservoir,  while  all  is  at  the 
room-temperature,  bring  the  mercury  to  the    wire  point  ''anil 
determine  the  difference  in  level  of  the  mercury  columns.   Re- 
cord the  room-temperature,  and  the  barometer-reading. 

(b)  After  having  lowered  the  mercury  on  the  bulb  side, 
surround  the  bulb  with  shaved  ice,  and  then  determine  the 


Q2  CONSTANT- VOLUME  AIR-THERMOMETER.  [42 

pressure  with  the  meniscus  at  the  wire  point.     The  tempera 
ture  may  be  taken  as  o°C. 

Melt  the  ice  with  warm  water,  and  then  make  a  series  of  de- 
terminations of  the  pressure  when  the  water  in  the  vessel  is 
successively  at  a  temperature  of  10°,  20°,  30°,  45°,  60°,  and 
8o°C.  ( approximately  )X 

Remove  the  water-bath,  substitute  a  steam-bath  in  its  place, 
and  make  another  determination.  The  temperature  of  the 
steam-bath  may  be  found  by  determining  the  boiling  point  of 
water  from  the  known  atmospheric  pressure  (see  Tables). 

Arrange  all  observations  in  tabular  form. 

(c)  Plot  on  coordinate  paper  the  results  of   (b),  using 
temperatures  as  abscissae  and  the  corresponding  pressures  as 
ordinates.     Draw  a  smooth  curve  which  will     best  represent 
the  average  position  of  the  points  of  the  plot. 

Calculate  the  mean  increase  of  pressure  per  degree  increase 
in  temperature  from  o°C.  to  ioo°C.,  and  divide  the  result  by 
the  pressure  at  o°C.,  using  values  taken  from  the  plot.  This 
is  the  temperature  coefficient  (ft)  of  pressure  of  a  gas. 

£U& 

Write  it(-is)a  decimal  and  find  its  reciprocal.  The  negative  of 
this  represents  what  point  on  the  absolute  scale  of  tempera- 
tures ? 

(d)  Write  an  equation  connecting  P0,  the  pressure  at  o°  ; 
P,  the  pressure  at  t°  ;  t ;  and  (3. 

Using  this  equation  and  the  pressure  obtained  in  (a),  cal- 
culate the  temperature  of  the  room,  thus  using  the  apparatus 
as  a  thermometer.  Compare  the  result  with  the  room  tem- 
perature as  read  from  a  mercury  thermometer. 

(e)  Show  from  your  results  how  the  pressure  of  the  gas 
varies  with  the  absolute  temperature,  the  volume  remaining 
constant. 


43]  VAPOR-PRESSURE    AND    VOLUME.  93 

43.      VAPOR-PRESSURE    AND    VOLUME. 
References. — Duff,  p.  227;  Millikan,  p.  152. 

The  purpose  of  this  experiment  is  to  study  the  relation  be- 
tween the  vapor-pressure  of  a  saturated  vapor  and  the  vol- 
ume of  the  vapor,  when  its  temperature  is  kept  constant. 

An  ordinary  barometer  tube,  about  a  meter  long  and  filled 
with  mercury,  is  inverted  in  a  cistern  of  mercury.  A  small 
amount  of  the  liquid,  whose  vapor-pressure-  is  to  be  studied, 
is  introduced  into  the  tube,  rising  to  the  top  of  the  tube  and 
vaporizing.  By  raising  or  lowering  the  tube  in  the  cistern, 
the  volume  of  the  vapor  can  be  changed.  The  corresponding 
vapor-pressure  is  found  by  determining  the  difference  between 
the  barometer-reading  and  the  height  of  the  mercury  column 
above  the  mercury  in  the  cistern. 

Experience  has  shown  that  the  saturated  vapor  above  the 
surface  of  a  liquid  exerts  a  pressure  which  depends  only  on 
the  nature  and  temperature  of  the  liquid.  It  should  follow 
from  this,  if  the  temperature  is  kept  constant,  that  the  vapor- 
pressure  of  any  given  liquid  is  independent  of  the  volume  of 
the  vapor,  as  long  as  the  vapor  remains  saturated. 

(a)  Unsaturated  and  Saturated  Vapors.  Fill  a  barometer 
tube  with  clean  mercury  to  within  I  cm.  of  the  end,  and  plac- 
ing the  thumb  over  the  end,  invert  the  tube  slowly  so  as  to 
make  the  large  air-bubble  pass  up  along  the  tube.  If  this  is 
done  a  number  of  times  and  along  different  sides  of  the  tube, 
most  of  the  air-bubbles  will  be  washed  out  of  the  tube. 
Now  fill  the  tube  brimming  full  and  invert  in  the  cistern,  tak- 
ing care  not  to  allow  any  air  to  enter.  Clamp  the  tube  in  a 
vertical  position  and  measure  the  height  of  the  mercury  col- 
umn above  the  surface  of  the  mercury  in  the  cistern.  Com- 
pare with  the  reading  of  the  laboratory  barometer,  and  if  the 
difference  is  greater  than  i  cm.,  remove  the  tube  and  fill  again 
more  carefully. 


94  VAPOR-PRESSURE  AND  VOLUME.  [43 

With  a  medicine  dropper  introduce  a  little  ether  into  the 
tube,  taking  care  not  to  force  any  air  in  with  it.  Observe  if 
the  ether  all  vaporizes  or  not;  note  the  height  of  the  mercury 
column ;  and  continue  to  add  more  and  more  ether  until  the 
mercury  stops  falling.  Finally  record  the  height  of  the  mer- 
cury column  and  the  length  of  the  vapor-column  above  it.  As- 
suming that  the  temperature  has  remained  equal  to  room-tem- 
perature, what  can  you  conclude  about  the  relative  pressures  of 
saturated  and  unsaturated  ether-vapor  at  the  same  tempera- 
ture? 

(b)  Vapor-Pressure  and  Volume.    Unclamp  the  tube  and 
gradually  lower  it  in  the  cistern,  20  cm.  at  a  time.     After 
waiting  a  minute  each  time  for  equilibrium  to  be  established 
between  the  vapor  and  the  liquid,  read  the  height  of  the  mer- 
cury column  and  the  length  of  the  vapor  column.     From  the 
height  of  the  mercury  column  and  the  barometer-reading  the 
vapor-pressure  can  be  found,  while  the  length  of  the  vapor - 
column  may  be  taken  to  represent  the  volume  of  the  vapor. 
Continue  until  all  of  the  vapor  is  condensed.     The  gas  which 
remains  is  air,  and  if  present  in  considerable     amount,     its 
pressure  will  constitute  an  appreciable  part  of  the  measured 
pressure,  especially  in  the  later  measurements. 

(c)  Mixture  of  Vapor  and  Gas.    With  the  medicine  drop- 
per force  more  air  into  the  tube  and  repeat  the  measurements 
of  (b). 

Remove  the  barometer  tube  from  the  cistern,  fill  again  with 
clean  mercury,  introduce  enough  air  to  cause  the  mercury  col- 
umn to  drop  20  cm.  or  so,  and  repeat  the  measurements  of  (&\ 

What  effect  does  the  presence  of  a  gas  in  the  vapor  have  on 
the  results? 

(e)  Vapor-Pressure  and  Nature  of  the  Liquid.    Repeat  (a) 
with  alcohol,  and  if  time  permits,  with  water  also.     Does  the 
vapor-pressure  depend  upon  the  nature  of  the  liquid? 

(f)  Plot  the  results  of  (a),  (b),  (c),  and  (d)  for  ether  on  a 


44]  VAPOR-PRESSURE    AND    TEMPERATURE.  95 -is 

single  sheet  of  coordinate  paper,  with  pressures  as  ordinates 
and  volumes  as  abscissae. 

From  the  results  and  the  curves  determine  (i)  how  the 
pressure  of  a  vapor  at  a  given  temperature  depends  upon  the 
degree  of  saturation,  (2)  how  the  pressure  of  a  saturated 
vapor  at  a  given  temperature  depends  upon  the  volume  of  the 
vapor,  and  (3)  whether  the  vapor  pressure  depends  upon  the 
nature  of  the  liquid  or  not. 


44.     VAPOR-PRESSURE  AND  TEMPERATURE. 
References. — Duff,  p.  227;  Millikan,  p.  152. 

The  object  of  this  experiment  is  to  study  the  relation  be- 
tween the  temperature  and  the  pressure  of  saturated  water- 
vapor.  The  method  employed  is  that  referred  to  in  Exp.  22 
as  the  "static"  method  of  determining  the  boiling  point  of  a 
liquid  at  different  pressures.  Two  barometer  tubes,  filled  with 
mercury,  are  inverted  and  mounted  side  by  side  in  a  vessel  of 
mercury.  One  of  the  tubes  contains,  above  the  mercury, 
water-vapor  with  an  excess  of  water  present,  while  the  other 
tube  is  left  to  be  used  as  a  barometer.  By  means  of  a  water- 
bath  surrounding  the  upper  half  of  the  tubes,  the  temperature 
of  the  water-vapor  can  be  brought  to  any  desired  point.  The 
bath  is  connected  to  a  heater  and  the  change  in  temperature  is 
brought  about  by  circulation.  The  pressure  of  the  saturated 
water-vapor  at  any  temperature  will  be  the  difference  between 
the  heights  of  the  mercury  columns  in  the  two  tubes. 

At  each  temperature  the  pressure  of  a  saturated  vapor  of  a 
given  liquid  has  a  definite  value  which  depends  on  the  tem- 
perature and  the  nature  of  the  liquid,  but  is  independent  of  the 
volume  of  the  vapor.  When  the  temperature  is  raised,  not 
only  is  the  vapor  heated  and  the  pressure  raised,  but  more  li- 
quid is  vaporized,  so  there  are  two  influences  tending  to  in- 
crease the  pressure  of  the  vapor.  The  purpose  of  the  present 


96  VAPOR-PRESSURE    AND    TEMPERATURE.  [44 

experiment  is  to  plot  the  curve  which  shows  how  rapidly  the 
vapor-pressure  increases  as  the  temperature  is  raised,  in  the 
case  of  water-vapor. 

(a)  Read  the  heights  of  the  mercury  columns  in  the  two 
tubes  for  ten  different  temperatures  between  room-temperature 
and  8o°C.   (approximately).     To  raise  the  temperature  about 
5  or  i o°  at  a  time,  heat  the  boiler  only  for  two  or  three  min- 
utes, then  remove  the  burner,  and  stir  the  water-bath  until 
a  uniform  temperature  prevails.    By  this  time  the  water-vapor 
inside  the  tube  will  have  reached  the  temperature  of  the  bath. 
In  taking  the  temperature-readings  hold  the  bulb  of  the  ther- 
mometer  slightly  below   the   center  of  the   space  filled   with 
water-vapor.     The  mercury-equivalent  of  the  column  of  water 
above  the  mercury  in  the  tube  containing  the  vapor  should  be 
taken  into  account  in  estimating  the  pressure. 

Always  wait  until  conditions  have  become  steady  before  tak- 
ing readings  at  a  new  temperature. 

(b)  While  the  temperature  is  falling  by  radiation,  take  as 
many  readings  of  both  mercury  columns    (at     intervals     of 
about  5°)  as  time  will  allow. 

(c)  Plot  a  curve  from  the  results  of  (a)  and  (b),  with  the 
pressures  of  the  saturated  water-vapor  as  ordinates  and  the 
temperatures  as  abscissae.    Draw  the  curve  so  that  it  will  rep- 
resent the  average  positions  of  all  the  plotted  points. 

(d)  Extend  the  curve  back  to  intersect  with  the  pressure 
ordinate  corresponding  to  o°C.     Assuming  that,   instead   of 
water-vapor,  you  are  given  a  perfect  gas  whose  pressure  at 
o°C,  is  the  same  as  that  of  the  saturated  water- vapor,  calculate 
what  the  pressure  of  the  gas  would  be  at  25°,  50°,  75°,  and 
ioo°C.,  its  mass  and  volume  being  kept  constant.     For  this 
purpose  it  will  be  convenient  to  use  the  law  expressing  the 
relation  between  the  pressure  and  the  absolute  temperature  of 
a  given  mass  of  a  perfect  gas  kept  at  constant  volume.     Plot 
the  results  in  a  curve,  and  compare  with  the  curve  obtained 
in   (<:)• 


45]  HYGROMETRY.  97 

Does  the  pressure  of  saturated  water-vapor  increase  with 
the  temperature  more  or  less  rapidly  than  does  the  pressure  of 
a  gas  kept  at  constant  volume? 

Would  the  results  be  different  if  the  volume  of  the  satu- 
rated vapor  were  kept  constant?  (See  Exp.  43.) 

Determine,  from  the  curve  in  (d),  the  boiling  point  of  water 
at  a  pressure  of  50  cm. 


45.     HYGROMETRY. 
References. — Duff,  p.  240;  Millikan,  p.  164;  Edser,  p.  240. 

In  this  experiment  the  dew-point  and  the  relative  and  abso- 
lute humidity  of  the  air  are  to  be  determined.  The  absolute 
humidity,  d,  is  the  density  of  the  water-vapor  present  in  the 
air,  and  is  usually  expressed  in  grams  per  cubic  meter.  The 
relative  humidity  is  the  ratio  of  the  amount  of  water-vapor 
actually  present  in  the  air  to  the  amount  required  to  saturate 
it  at  the  same  temperature^  the  latter  quantity  being  the  max- 
imum amount  of  water-vapor  that  can  be  held  in  suspension 
at  that  temperature.  The  relative  humidity  is  therefore  equal 
to  d/D,  where  D  is  the  maximum  density  of  the  water-vapor 
at  the  given  temperature.  The  dew-point  is  the  temperature 
at  which  the  amount  of  water  actually  present  in  the  air  would 
saturate  it,  that  is,  the  temperature  to  which  the  air  must  be 
lowered  before  the  condensation  of  water  will  begin.  The 
pressure  of  water-vapor  is  the  pressure  which  it  would  exert 
by  itself  if  there  were  no  air  present  in  the  space  considered. 
By  Dalton's  law  this  is  the  pressure  it  actually  does  exert  when 
mixed  with  air.  In  a  given  volume  the  mass  of  vapor  is  pro- 
portional to  the  pressure,  so  that  the  relative  humidity  is  equal 
to  the  ratio  of  the  pressure  p  of  the  water-vapor  in  the  air  to 
the  pressure  P  of  saturated  water-vapor  at  that  temperature; 
that  is,  relative  humidity  is  equal  to  p/P. 


98  HYGROMETRY.  [45 

(I)     Regnaulfs  Hygrometer. 

(a)  Partially  fill  one  of  the  hygrometer  tubes  with  ether, 
and  place  a  thermometer  in  the  liquid.     Force  a  current  of 
air  through  the  ether  with  a  bicycle  pump.    The  rapid  evapor- 
ation of  the  liquid  causes  the  temperature  to  fall.  When  the 
tube  and  the  air  immediately  above  it  are  cooled  to  the  dew- 
point,  moisture  appears  on  the  tube,  this  being  detected  more 
easily  by  comparison  with  the  other  tube.    Note  the  tempera- 
ture at  which  the  dew  begins  to  form.     Allow    the    tube  to 
become  warm  and  record  the  temperature  at  which  the  dew 
disappears.     Take  the  mean  of  these  two  as  the  dew-point. 
Make  three  such  determinations  of  the  dew-point. 

(b)  From  the  Tables   find     the  pressure  of     saturated 
water-vapor  at  the  dew-point  and  also  at  the  temperature  of 
the  room,  and  calculate  the  relative  humidity.     The  absolute 
humidity  may  be  found  by  multiplying  the  relative  humidity 
by  D,  the  number  of  grams  of  saturated  water-vapor  in  a  cubic 
meter  of  air  at  the  room-temperature   (see  the  Tables). 

II.  Wet-  and  Dry-bulb  Hygrometer,  or  Auguste's  Psy- 
chrometer. 

In  the  wet-  and  dry-bulb  hygrometer,  one  bulb  is  covered 
with  wicking  which  dips  into  water,  so  that  the  bulb  is  cooled 
by  evaporation.  Swing  the  hygrometer  back  and  forth  in  the 
air  so  as  to  increase  the  circulation  of  air  about  the  wet  bulb. 
After  the  two  thermometers  come  to  constant  temperatures, 
record  the  temperature  t  of  the  dry  bulb,  and  the  tempera- 
ture ^  of  the  wet  bulb.  Read  the  barometer.  The  following 
empirical  formula  may  then  be  used: 

/>  =  />!  —  0.0008  b  (f— *!>, 

where  p  is  the  pressure  of  water- vapor  present  in  the  atmos- 
phere and  the  value  of  which  is  to  be  found;  p^  the  pressure 
of  saturated  vapor  at  the  (temperature  of  the  wet-bulb  (ob- 


46]  DENSITY   OF   THE   AIR    BY   THE   BARODEIK.  99 

tained  from  the  Tables)  ;  and  b  is  the  barometric  pressure,  all 
being  expressed  in  millimeters  of  mercury.  Find  the  pres- 
sure P  of  saturated  water-vapor  at  the  room-temperature 
from  the  Tables,  and  calculate  the  relative  humidity.  Find 
then  the  absolute  humidity  as  in  (fr).  From  the  Tables  and 
the  readings  of  the  wet-  and  dry-bulb  hygrometer^  find  the 
dew-point. 

Compare  the  values  obtained  in  I  and  II  for  the  humidity 
and  the  dew-point. 


46.    DENSITY  OF  THE  AIR  BY  THE  BARODEIK. 

The  barodeik  is  an  ordinary  balance,  having  a  hermetically 
sealed  flask  suspended  from  one  scale-pan,  and  from  the  other 
(as  a  counterpoise)  a  glass  plate  so  chosen  as  to  have  a  surface 
about  equal  to  the  exterior  surface  of  the  flask.  The  reading 
of  the  balance-pointer  on  a  properly  graduated  scale  gives 
the  density  of  the  surrounding  air. 

I.  To  find  the  difference  between  the  barodeik  reading  and 
the  true  density  of  the  air. 

(a)  Set  and  read  the  barometer  with  great  care.  Read  the 
wet-  and  dry-bulb  hygrometer.    From  the  Tables  calculate  the 
dew-point  and  also  the  pressure  of  the  water-vapor  in  the  air. 
Remember  that  "dew-point"  means  the  temperature  at  which 
the  water- vapor  now  in  the  air  would  be  saturated,  or  the 
temperature  at  which  the  existing     pressure  of  the     water- 
vapor  in  the  air  would  be  the  maximum  pressure. 

(b)  From  (a)  calculate  the  density  of  the  air.     The  mass 
of  one  cu.  cm.  of  dry  air,  at  o°C.,  and  76  cm.     pressure,  is 
0.001293  grams.     The  mass  of  the  same  volume  of  water- 
vapor,  under  the  same  conditions,  is  5/8  as  much.     Then,  if 
H  be  the  barometric  height,  f  the  pressure  of  water-vapor,  and 


IOO  DENSITY  OF  THE  AIR  BY  THE  BARODEIK.  [46 

t  the  temperature,  the  mass  of  dry  air  in  one  cu.  cm.  of  moist 
air  is  by  the  general  gas  law,  PV  =  RmT, 

i       H  -f 
Jf1  =  0.001293  I+g,       ?6     , 

where  a  is  the  coefficient  of  expansion  of  a  gas.    The  mass  of 
water-vapor  in  the  same  volume  is 


293 IT*  7-6- 

The  sum  of  these  two  is  the  required  density.     (Deschanel, 
p.  400.) 

(c)  Read  the  barodeik.     Do  not  touch  the  instrument,  but, 
by  moving  the  hand  near  the  flask,  set  up  a  small  vibration ; 
then  close  the  case,  and  determine  the  resting-point  of  the 
pointer,  which  is  the  density  of  the  air  as  indicated  .by  the  in- 
strument. 

(d)  Record  the  difference  between  the  reading  thus  ob- 
tained and  the  true  density  found  in   (b)  ;  prefix  the  proper 
sign,  so  that,  when  added  algebraically  to  the  observed  read- 
ing, it  will  give  the  true  density  of  the  air.    This  is  the  abso- 
lute correction  for  the  scale-division  to  which  it  applies. 

II.     Relative  Calibration  of  the  Barodeik  Scale. 

(a)  Read  the  instrument  as  in  I   (c).     Repeat  with  the 
rider  at  division  2  to  the  right  of  the  center  of  the  scale,  which 
is  equivalent  to  adding  2  mg.  to  the  right-hand  pan  of  the 
balance;  then  use  the  rider  in  the  corresponding  position  on 
the  left-hand  side. 

(b)  Repeat  the  readings  with  the  rider  at  division  5,  first 
on  the  right-hand  side,  then  on  the  left-hand  side. 

(c)  Using  the  exterior  volumes  of  flask  and  plate  as  given 
on  the  instrument,  calculate  the  changes   in   the  density  of 
the  air  which  would  produce  the  same  effects  on  the  instru- 
ment as  the  putting  of  the  separate  masses  on  the  right  pan, 


47]  COEFFICIENT  OF  FRICTION.  101 

and  on  the  left  pan.  From  these  results  construct  a  table  of 
corrections,  with  the  proper  signs,  for  the  different  resting- 
points  observed.  Note  that  this  is  a  relative  calibration ;  that 
is,  it  gives  the  corrections  to  be  applied  to  certain  readings,  as 
compared  with  one  reading  (namely,  that  when  no  weights 
were  used)  which  is  assumed  correct. 

(d)  In  part  I  the  absolute  correction  for  a  certain  reading 
was  found.  That  reading  was  the  same  as,  or  not  far  from,  the 
one  assumed  correct  above,  so  the  same  absolute  correction 
may  be  applied  to  the  latter.     By  means  of  this,  convert  the 
table  of  relative  corrections,   (c),  into  a  table     of     absolute 
corrections  .  This  completes  the  absolute    calibration     of  the 
instrument. 

(e)  Plot  on  coordinate  paper  the  readings  of  the  barodeik 
scale  as  abscissae  and  the  relative  corrections  of  (c)  as  ordi- 
nates?  but  on  a  much  larger  scale.     Show  how  the  curve  can 
be  made  to  indicate  absolute  corrections  instead  of  relative, 
by  moving  the  horizontal  axis  of  reference  up  or  down  by 
a  proper  amount.     This  converts  it  into  an  absolute  calibra- 
tion curve  for  the  instrument,  enabling  one  to  find  the  density 
of  the  air  at  any  time  by  merely  reading  the  resting-point  of 
the  pointer. 

47.    COEFFICIENT  OF  FRICTION. 

Reference.— Duff,  p.  95. 

When  one  body  is  caused  to  slide  over  the  surface  of 
another,  the  force  which  is  brought  into  play  to  oppose  the 
motion  is  called  "friction."  This  force  is  parallel  to  the  sur- 
face and  opposite  in  direction  to  the  motion.  When  the 
sliding  body  is  on  a  level  plane,  the  normal  force  is  equal  to 
the  weight  of  the  body;  when  on  an  inclined  plane  it  is 
equal  to  the  component  of  the  body's  weight  normal  to  the 
plane.  In  either  case  the  force  of  friction  is  equal  and  oppo- 


102  COEFFICIENT    OF    FRICTION.  [47 

site  to  the  force  necessary  just  to  produce  motion  (starting 
friction),  or  to  keep  the  body  moving  at  constant  speed  (mov- 
ing friction).  If  P  is  the  force  between  the  two  surfaces  and 
normal  to  them,  and  F  is  the  force  of  friction,  the  ratio 


is  called  the  coefficient  of  starting  or  moving  friction,  as  the 
case  may  be,  and  is  usually  denoted  by  the  Greek  letter  /x. 
By  measuring  these  forces  and  calculating  their  ratio  the 
coefficient  may  be  determined.  A  second  method  of  deter- 
mining the  coefficient  of  friction  is  to  vary  the  inclination  of 
the  plane  until  the  body  by  its  weight  just  begins  to  move 
(starting  friction)  or  moves  down  the  plane  with  constant 
speed  (moving  friction).  If  the  angle  of  inclination  at 
which  this  occurs  is  i,  it  can  be  shown  that  the  coefficient  of 
friction  is  equal  to  tan  i. 

(a)  The  coefficient  of  friction  is  to  be     found     between 
blocks  and  the  surface  of  a  plane  whose  inclination     can  be 
varied.    Take  one  of  the  blocks  and  weigh  it.    Determine  the 
force  of  starting  friction  and  also  of  moving  friction  on  a  level 
surface  by  applying  forces  to  it  by  means  of  the  shot-bucket 
and  string  and  pulley.    Calculate  the  coefficient  of  friction  for 
the  two  cases. 

(b)  Determine  the  coefficient  of  friction   for     the     same 
block  and  surface  from  the  tangent  of  the  angle  obtained  by 
varying  the  inclination  of  the  plane  until    (i)    motion  com- 
mences, and   (2)  motion  continues  at  constant  speed. 

(c)  Set  the  plane  at  the  angle  giving  constant  speed  down 
the  plane,  and  find  the  force  that  will  cause  the  block  to  move 
up  the  plane  at  constant  speed.     Calculate  the  coefficient  of 
friction. 

(d)  Set  the  plane  at  an  angle  of  30°  and  find  the  force 
necessary  to  move  the  block  up  the  plane  at  constant  speed, 


48]  CONSERVATION   OF   MOMENTUM.  1 03 

and  then,  if  possible,  the  force  necessary  to  make  it  move 
down  the  plane  at  constant  speed.  Then,  by  calculating  the 
force  perpendicular  to  the  plane,  find  the  coefficient  of  fric- 
tion. If  this  process  is  not  entirely  clear,  repeat  with  the  plane 
at  an  angle  of  60°.  N 

(e)  Repeat  (a),  for  starting  friction>  having  the  block 
"loaded"  by  placing  a  known  mass  on  top  of  it.  Compare 
the  coefficient  of  friction  found  with  that  found  in  (a). 

(/)  Take  a  block  having  three  or  more  surfaces  of  differ- 
ent areas  but  of  the  same  smoothness,  and  determine  (by  any 
method)  the  force  of  friction  as  the  block  slides  or  is  moved 
successively  on  the  three  surfaces. 

(g)  Take  a  block  with  surfaces  of  different  degrees  of 
smoothness,  and  determine  the  coefficient  of  starting  friction 
for  two  or  more  sides. 

(h)  Compare  the  results  obtained  from  (a),  (b),  (c), 
(d),  and  (e),  stating  your  conclusions.  What  do  you  con- 
clude from  (/)  ?  From  (g)  ?  Upon  what  does  the  friction 
between  two  surfaces  depend? 

48.     CONSERVATION  OF  MOMENTUM.     COEF- 
FICIENT OF  RESTITUTION. 

:    '  K 

Reference. — Millikan,  p.  58. 

In  any  system  of  bodies,  which  is  not  acted  upon  by  outside 
forces  and  in  which  the  several  bodies  may  be  moving  with 
different  velocities  and  in  different  directions  with  frequent 
collisions,  the  vector  sum  of  the  momenta  remains  constant. 
This  is  known  as  the  Law  of  Conservation  of  Momen- 
tum. In  our  present  study  the  number  of  bodies  will  be  lim- 
ited to  two  and  velocities  restricted  to  the  same  straight  line, 
the  collisions  taking  place  centrally.  Let  us  suppose  that  we 
have  two  bodies  A  and  B,  suspended  by  strings  so  that  they 
hang  in  contact  when  at  rest  Let  A  be  drawn  aside  and  then 


IO4  CONSERVATION    OF    MOMENTUM.  [48 

released.  At  the  lowest  point  of  its  swing  it  strikes  the  ball 
B.  Let  m±  be  the  mass  of  A  and  «t  its  velocity  just  as  it 
strikes  B.  Its  momentum  then  at  this  instant  is  mvu^  The 
ball  B  will  at  once  start  off  with  a  velocity  ^2,  say,  and  a  mo- 
mentum m2z>2,  if  m2  is  its  mass.  The  ball  A  may  continue  on 
with  a  diminished  velocity,  v^  ;  or  remain  at  rest,  if  it  loses  all  of 
its  momentum ;  or  it  may  rebound,  in  which  case  v1  is  negative. 
After  impact  the  two  balls  will  move  away  from  each  other 
with  a  relative  velocity  which  is  greater  the  greater  their  elas- 
ticity. The  elasticity  is  taken  into  account  in  a  factor  called 
the  "coefficient  of  restitution."  The  coefficient  of  restitution 
is  numerically  equal  to  the  ratio  of  the  relative  velocities  with 
which  the  bodies  move  apart  after  impact  to  that  with  which 
they  approached  each  other  before  impact^  i.  e.,  it  is  given  by 
the  equation, 

V  --  V 

(1)  *  =  ^-  — l, 

«t   —   U, 

where  the  velocities  before  impact,  wx  and  u2,  and  the  veloc- 
ities after  impact,  v±  and  vz,  are  all  in  the  same  straight  line. 
One  or  more  of  the  velocities  may  be  negative,  or  the  par- 
ticular value  of  a  velocity  may  be  zero,  as  in  the  case  just 
outlined  for  the  two  balls  where  «2  =  o  since  the  second  ball 
was  at  rest  before  the  impact.  The  value  of  e  always  lies  be- 
tween zero  and  unity.  For  "perfectly  elastic"  bodies  e  =  i, 
but  for  all  actual  bodies  £<i.  For  inelastic  bodies  e  =  o.  In 
any  case,  whether  the  bodies  are  elastic  or  inelastic,  the  con- 
servation of  momentum  holds,  i.  e., 

(2)  w^!  -f-  m2u2  =  m^Vi  -f-  f»2?'2. 

This  may  be  verified  by  determining  the  masses  and  the  ve- 
locities. The  purpose  of  the  experiment  is  to  verify  relation 
(2)  in  the  simple  case  just  outlined  for  tzuo  suspended  balls. 
In  order  to  find  the  velocity  of  a  suspended  ball  as  it  col- 
lides, or  just  after  collision,  we  make  use  of  the  fact  that  the 


48]  CONSERVATION    OF    MOMENTUM.  IO5 

velocity  is  the  same  as  that  which  the  ball  would  have  ac- 
quired if  it  had  fallen  the  same  vertical  distance  that  it  has  de- 
scended in  its  swing  before  collision,  or  that  it  has  risen  in  its 
swing  after  collision,  as  the  case  may  be.  Let  the  height  be 
h ;  then,  as  the  case  may  be,  u  or  v  equals  \/2gh  where  g  is 
the  acceleration  due  to  gravity.  If  the  angle  of  the  half  swing 
is  a  and  the  length  of  the  pendulum  is  /,  we  have, 

(3)  u,  or  v,  =  \/2gl  (i  —  cos  a). 

(a)  The  numbers  on  the  circular  scale,  at  the  bottom  of 
the  frame  from  which  the  balls  are  suspended,  represent  de- 
grees of  arc.    First  use  the  two  large  ivory  balls,  and  see  that 
they  are  adjusted  so  as  to  hang  fully  in  contact  and  so  that 
their  centers  are  in  line.     Record  the  zero-reading  for  each 
ball,  the  other  one  being  drawn  aside.     Then  draw  one  aside 
through  about  10°  or  15°  and  fix  it  in  position  with  a  thread. 
Record  the  reading.     Release  it  by  burning  the  thread.     Note 
carefully  the  extremity  of  the  swing  of  each  ball  after  impact. 
This  can  be  done  by  placing  a  slider  in  the  position  for  each 
ball.     Several  trials  will  be  necessary  to  accurately  determine 
these  points.     From  these   and  the  zero-readings,  the  arcs  of 
the  swings  are  found,  and  then  by  measuring  the  length  of 
the  cord   (to  the  center  of  the  ball)  the  velocities  ult  vlt  and 
v2  can  be  determined.     Repeat  for  two  other  starting  points 
from  which  the  ball  is  released.     Determine  the  masses  of  the 
balls,  and  then  calculate  for  the  three  cases  the  momentum  of 
the  system  before  and  after  impact. 

(b)  Use  one  large  ivory  ball  and  one  small  one,  and  repeat 
for  one  or  two  starting  points,  releasing  the  large  ball. 

(c)  Repeat    (&),   reversing  the   process  by   releasing  the 
small  ball. 

(d)  Use  two  lead  balls  and  repeat  for  one  or  two  starting 
points,  or  place  a  layer  of  paraffine  on  each  large  ivory  ball  on 
adjacent  sides  and  use  them  as  inelastic  balls. 


io6  YOUNG'S  MODULUS  BY  STRETCHING.  [49 

(e)  Calculate  the  coefficient  of  restitution  for  all  the  cases 
above.  Does  it  appear  to  depend  principally  on  the  size  of 
the  balls  or  on  the  material? 

For  the  same  cases  calculate  the  percentage  difference  in 
the  momentum  before  and  after  impact.  Does  the  "Conserva- 
tion of  Momentum"  appear  to  hold  equally  well  in  all  cases  ? 

Calculate  the  percentage  of  loss  of  kinetic  energy  for  each 
case.  What  becomes  of  the  energy  apparently  lost?  Is  the 
loss  greater  in  the  more  or  in  the  less  elastic  bodies? 

49.     YOUNG'S  MODULUS  BY  STRETCHING. 

References. — Duff,  pp.   119,   121;   Millikan,   p.   65. 

Hooke's  Law  states  that  in  elastic  bodies,  within  their  elas- 
tic limits,  the  strain  or  deformation  produced  is  proportional 
to  the  change  in  the  stress  or  distorting  force.  In  particular  it 
states  that  if  different  forces  be  applied  to  a  wire,  e.  g.,  by 
suspending  it  and  hanging  masses  from  it,  the  amount  of 
stretching  will  be  (within  certain  limits)  proportional  to  the 
applied  force.  For  a  wire  of  any  given  material  the  ratio  of 
the  change  in  stress  per  unit  area  of  cross-section  to  the  in- 
crease in  length  per  unit  length  is  known  as  Young's  modulus. 
If  P  is  the  additional  force  applied  to  a  wire  of  length  L  and 
cross-section  a,  and  /  is  the  elongation  produced,  the  value  of 
t^ie  ratio  is 

P       J  PL 

— =-  ,  ,     or     — =—  . 
a       L  al 

It  is  approximately  a  constant  for  any  given  material,  thus 
verifying  Hooke's  law.  The  constant  has  widely  varying  val- 
ues, however,  for  different  materials. 

To  determine  Young's  modulus  for  any  metal,  a  wire  made 
of  that  metal  is  held  vertically  between  two  clamps.  To  the 
lower  clamp  C  is  attached  the  end  of  a  rod  whose  upper  end 
is  loosely  held  in  a  support.  From  the  lower  end  of  the  wire 


49]  YOUNG'S  MODULUS  BY  STRETCHING.  107 

or  the  clamp  C,  masses  may  be  suspended  and  the  wire 
stretched.  Above  the  upper  end  of  the  rod  is  a  micrometer 
screw  with  a  divided  head  so  that  small  fractions  of  a  turn 
may  be  read.  In  one  form  of  the  apparatus,  when  the  screw 
comes  in  contact  with  the  rod  an  electric  bell  rings.  If  the 
wire  is  then  stretched,  the  screw  must  be  advanced  again  be- 
fore ringing  will  occur.  In  this  way  the  change  in  length  of 
the  wire  between  the  two  clamps  is  readily  determined,  if  the 
pitch  of  the  screw  is  known.  In  the  other  form  of  the  ap- 
paratus a  small  spirit  level  is  used,  one  end  of  which  is  at- 
tached to  the  upper  end  of  the  rod  and  the  other  end  rests  on 
the  point  of  the  micrometer  screw.  If  the  wire  is  stretched 
the  screw  must  be  turned  back  before  the  spirit  level  will 
again  read  as  before.  The  amount  of  the  stretching  can  thus 
be  determined  as  with  the  other  apparatus. 

(a)  Hang  first  a  large  enough  mass  on  the  wire  to  insure 
that  it  is  straight  at  the  beginning.     Make  a  setting  with  the 
screw  and  read  it  to  o.oi  mm.  or  less.     Then     increase    the 
load  by  adding  500  gms.  at  a  time  (each  time     reading  the 
screw),  until  the  wire  carries  a  load  of  about  3000  gm.,  or 
until  the  elastic  limit  is  approached. 

(b)  Take  the  masses  off,  500  gm.  at  a  timej     reading  the 
screw  each  time. 

(c)  Repeat  (a)  and  (b)  at  least  once.    Record  the  obser- 
vations in  tabulated  form.     Calculate  and  record  the  elonga- 
tion produced  by  each  500  gm. 

(d)  Measure  the  length  of  the  wire  between  the  clamps, 
and  also  the  diameter  of  the  wire.     In  measuring  the  latter, 
apply  the  caliper  to  four  or  five  different  points  along  the  wire. 

(f)  From  the  data  for  each  wire  calculate  the  mean  elon- 
gation of  that  wire  produced  by  a  stretching  force  of  500 
grams-weight.  Expressing  the  force  in  dynes  and  the  lengths 
in  cm,,  calculate  Young's  modulus  for  each  of  the  wires,  and 


IO8  HOOKAS  LAW  FOR  TWISTING  [50 

compare.     Should  the  result  be  independent  of  the  diameter 
of  the  wire? 

What  evidence  does  the  experiment  give  for  the  verification 
of  Hooke's  law?  If  there  is  any  variation  from  the  law,  as- 
sign a  reason  if  you  can. 


50.   HOOKE'S  LAW  FOR  TWISTING.  COEFFICIENT 
OF  RIGIDITY. 

References. — Duff,   pp.    117,    121;    Millikan,   p.   71. 

If  a  cylindrical  wire  or  rod  be  fixed  at  one  end,  and  the  free 
end  be  twisted  about  the  axis  of  the  wire,  no  change  of  vol- 
ume will  occur,  but  the  strain  in  the  wire  is  found  to  be  one 
of  shape  or  form  only,  a  shearing  strain.  The  tendency  which 
the  wire  has  to  recover  from  this  strain  is  called  elasticity 
of  form.  For  a  wire  of  given  material,  length  and  diameter, 
the  force-moment  producing  the  twisting  is  found  to  be 
(within  certain  limits)  proportional  to  the  angle  of  twist. 
This  statement  may  be  deduced  mathematically  from  Hooke's 
Law  which  states  that  in  elastic  bodies  (within  their  elastic 
limits)  the  strain  or  deformation  produced  is  proportional  to 
the  stress  or  distorting  force.  The  mathematical  reasoning 
establishing  the  relation  between  the  angle  of  twist,  the  force- 
moment  producing  the  twisting,  and  the  material,  length,  and 
radius  of  the  wire  is  not  simple,  involving  integral  calculus. 
The  object  of  this  experiment  is  to  establish  the  relation  ex- 
perimentally. 

If  M  is  the  moment  of  the  twisting  force  <f>  the  angle  of 
twist  in  radians,  /  the  length  of  the  rod,  and  r  its  radius,  we 
may  write 

2MI  2MI 

(i)  <f>  = ;,     or     n= — -j—  . 

Trnr  Trfir* 

In  the  above  equation,  n  is  constant  for  a  given  material  and 


50]  HOOKERS  LAW  FOR  TWISTING.  IOO, 

is  called  the  "coefficient  of  rigidity,"  or  sometimes  the  "mod- 
ulus of  torsion." 

The  apparatus  consists  of  two  heavy  table-clamps,  one  of 
them  carrying  a  wheel  about  a  half-foot  in  diameter.  In  the 
hub  of  the  wheel  is  a  socket  in  which  the  rod  to  be  tested  is 
centered  and  rigidly  fastened.  The  other  end  of  the  rod  is 
held  in  a  similar  socket  mounted  in  the  other  clamp.  A  scale- 
pan,  attached  to  the  rim  of  the  wheel,  is  for  the  load.  Two 
smaller  clamps,  supporting  graduated  arcs,  are  placed  in  po- 
sition at  desired  points  along  the  rod.  A  metal  pointer, 
clamped  to  the  rod  under  each  of  the  arcs,  provides  a  way  for 
determining  the  relative  twist  in  the  rod  between  the  two 
clamps.  For  testing,  four  rods  are  provided,  two  of  the  same 
diameter  but  different  substance,  and  two  of  the  same  sub- 
stance but  different  diameter. 

(a)  Set  the  rod  of  smaller  diameter  in     place,     clamped 
firmly  at  both  ends  to  prevent  slipping.     Place  the     pointers 
exactly  20  cm.  apart  and  adjust  the  graduated  arcs  in  such  a 
position  with  reference  to  the  pointers  as  to  avoid  errors  due 
to  parallax  in  making  the  readings.     Set  both  pointers  at  the 
zero  marks.  Place  a  2OO-gm.  mass  in  the  scale-pan  and  record 
the  positions  of  the  two  pointers.    The  twist  of  the  rod  between 
them  is  measured  by  the  difference  between  the  readings  of 
the  pointers. 

(b)  Repeat   (a),  adding  masses  to  the     pan,     preferably 
200  gm.  at  a  time  up  to  about  1.5  kg.,  or  until  the  "limit  of 
elasticity"  of  the  rod  is  reached.     Whenever     this     limit  is 
passed  the  rod  will  fail  to  untwist  completely  upon  the  removal 
of  the  masses  in  the  pan.    Record,  in  tabular  form,  the  masses 
tised,  the  corresponding  angle  of  twist,  and  the  increase  in  the 
angle  for  each  2OO-gm.  mass  added.    Measure  the  diameter  of 
the  wheel^  and  the  diameter  of  the  rod,  the  latter  with  great 
care. 


IIO          FRICTION    BRAKE.     POWER   SUPPLIED  BY   A   MOTOR.          [51 

(c)  Repeat    (a)   and    (&)    with  the  pointers  adjusted  to 
include  lengths  of  40  cm.  and  80  cm.  of  the  rod. 

(d)  Replace  the  rod  by  one  of  the  same  substance  but  dif- 
ferent diameter.     Measure  the  diameter  as  in  (b),  taking  ten 
or  more  readings  but  using  only  the  five  smallest  of  them  in 
averaging  for  a  mean  value.     Repeat  the  measurments  of  (c) 
for  a  length  of  this  rod  equal  to  80  cm. 

(e)  Repeat  (d)  with  a  rod  of  different  substance,  but  hav- 
ing the  same  length  and  radius  as  that  used  in  (d). 

(/)  From  your  results  show  how  the  angle  of  twist  varies 
with  the  twisting  moment,  with  the  length  of  the  rod,  and  with 
its  radius. 

Expressing  all  the  quantities  in  equation  (i)  in  the  units  of 
the  absolute  C.  G.  S.  system,  calculate  the  coefficient  of  rig- 
idity for  each  of  the  cases  above.  Note  if  its  value  is  depen- 
dent only  on  the  substance,  or  not.  Point  out  how  the  data  af- 
ford a  verification  of  Hooke's  law. 

If  the  radius  of  the  wire  be  measured  to  an  accuracy  of 
o.oi  mm.,  with  what  accuracy  should  the  length  be  measured  in 
order  that  the  result  may  be  affected  to  the  same  degree  by 
both? 


51.     FRICTION  BRAKE.    POWER  SUPPLIED  BY  A 

MOTOR. 

Reference. — Watson,  p.   116. 

The  object  of  this  experiment  is  to  measure,  by  means  of 
a  friction  brake,  the  power  delivered  by  an  electric  motor,  and 
to  study  the  effect  of  altering  the  friction  of  the  different 
parts.  An  electric  motor,  a  bank  of  incandescent  lamps 
arranged  in  parallel,  and  a  key  are  connected  in  series  with 
the  no- volt  power-circuit.  The  circuit  is  made  by  pressing 
the  key.  The  resistance  can  be  decreased  by  introducing  more 
lamps  into  the  circuit.  A  Prony  brake  is  used.  The  Prony 


5i ]        FRICTION  BRAKE.   POWER  SUPPLIED  BY  A  MOTOR.        in 

brake  consists  of  a  lever,  one  end  of  which  is  bound  around 
a  revolving  shaft  in  such  a  way  that  the  friction  produced 
will  tend  to  rotate  the  lever  in  the  direction  in  which  the 
shaft  revolves.  This  tendency  to  rotate  is  balanced  by  a 
spring  balance  acting  at  right  angles  to  the  lever,  or  by  the 
weight  of  masses  hung  from  the  lever.  If  P  is  the  force  in 
dynes  acting  on  the  lever  to  prevent  rotation,  and  L  the 
distance  from  the  line  of  P  to  the  center  of  the  shaft,  the 
power  absorbed  by  the  brake,  or  the  work  per  second^  will 
be  2-rrLnP,  where  n  is  the  number  of  revolutions  of  the  shaft 
per  second. 

(a)  Suspend  a  spring  balance  from  the  iron  stand,  and 
then  attach  it  below  to  the  lever  of  the  brake  so  that,  when 
the  motor  is  running,  the  balance  will  oppose  any  tendency 
of  the  brake  to  rotate.     Note  the  reading  of  the  balance  when 
the  motor  is  not  running.     Then  start  the  motor  by  gradually 
decreasing   the   resistance  given   by   the   incandescent   lamps, 
and,  with  the  motor  running  at  less  than  full  speed,  tighten 
the  belt  connecting     the  motor     to  the  shaft  of  the  brake. 
Allow  the  motor  to  run  at  full  speed  with  the  belt  taut,  and 
record  the  number  of  revolutions  of  the  shaft  in  three  min- 
utes, as  given  by  the  speed  counter.     Note  the  reading  of  the 
spring  balance  while  the  shaft  is  rotating.     Take  two  more 
readings  with  the  balance  at  different  points  along  the  lever. 

(b)  Tighten  the  screws  which  bind  the  wooden  blocks  of 
the  brake  against  the  shaft,  and  take  measurements  with  three 
different  lever  arms.     Note  if  the  lamps  grow  brighter  when 
the  friction  is  increased.     If  so,  what  can  be  said  about  the 
dependence  of  the  power  consumed  by  a  motor  on  the  load? 
The  effect  may  also  be  observed  by  tightening  the  belt  con- 
necting motor  and  brake-shaft. 

(c)  Calculate  the  power  delivered  by  the  motor  for  each 
of  the  six  measurements.     In  what  units  is  the  power  ex- 
pressed, if  the  force  of  the  balances  is  in  dynes  and  the  lever 


112  ABSORPTION   AND  RADIATION.  \$2 

arm  in  centimeters  ?  Reduce  the  results  to  horse-power.  If 
you  know  the  method  by  which  electrical  power  is  computed, 
show  how  the  efficiency  of  the  motor  may  be  calculated. 

(d)  Disconnect  the  friction  brake,  attach  the  spring  bal- 
ances to  a  cord,  and  hold  or  suspend  them  above  the  motor 
so  that  they  will  pull  in  parallel  lines,  thereby  pressing  the 
cord  against  half  of  the  periphery  of  the  motor-wheel.  Allow- 
ing the  motor  to  run  at  moderate  speed,  record  the  difference 
in  the  readings  of  the  two  spring  balances  as  the  cord  presses 
against  the  wheel. 

(e)  Repeat  (d)  for  the  other  pulley- wheel  on  the  motor- 
shaft,  exerting  as  nearly  as  possible  the  same  tension  as  be- 
fore.    Measure  the  diameter  of  each  of  the  wheels  and  see 
what  relation  exists  between  the  friction  and  the     radius     of 
the  wheels f  the  angle  of  contact  being  the  same  in  the  two 
cc.ses. 


52.    ABSORPTION  AND  RADIATION. 
References. — Duff,  p.  253;  Edser,  p.  436. 

It  is  well  known  that  a  dull  black  surface  absorbs  light 
more  readily  than  a  white  or  light-colored  one.  This  is 
shown  by  the  difficulty  in  illuminating  a  photographic  dark 
room  or  a  room  with  dark-colored  hangings.  The 
purpose  of  this  experiment  is  to  see  whether  the  relations 
which  hold  for  light  apply  also  to  the  vibrations  of  longer 
period  which  are  manifest  to  our  senses  only  through  the 
sensation  of  heat.  That  is,  it  is  proposed  to  study  the  rate 
of  absorption  of  heat  by  black  and  by  polished  surfaces,  and 
also  the  rate  at  which  heat  is  radiated  by  these  surfaces  to  a 
colder  body. 

(a)  A  box  lined  with  tin  has  an  opening  in  the  side  in 
which  three  thermometers  may  be  set  and  read  from  the 
outside  of  the  box.  The  bulb  of  one  of  the  thermometers  is 


53]  RATIO  OF  THE  TWO  SPECIFIC  HEATS  OF  AIR.          113 

bare,  another  is  silvered,  and  the  third  is  coated  with  lamp- 
black. All  three  thermometers  should,  initially,  register  the 
temperature  of  the  room.  Record  the  room-temperature.  Heat 
water  to  boiling  in  a  kettle  and  pour  into  the  vessel  in  the  box, 
arranging  this  so  that  the  steam  will  not  reach  the  thermom- 
eter-bulbs and  condense  on  them.  Record  the  readings  of  all 
three  thermometers  each  minute  until  a  steady  temperature  is 
reached.  Then  at  an  even  minute  remove  the  hot  water  and 
continue  the  readings  till  the  thermometers  again  register 
the  temperature  of  the  room. 

(b)  Make  a  good  freezing  mixture  in  a  large  beaker,  and 
place  this  in  the  box  close  to  the     thermometer-bulbs,     the 
thermometers  being  equally   distant   from   the   freezing  mix- 
ture.    Read   the  temperatures   each   minute  until  they   cease 
to  fall.     Remove  the  freezing  mixture  and  read  the  thermom- 
eters as  they  return  to  room  temperature. 

(c)  Plot  the  results  of  (a)  and  (b)  on  coordinate  paper, 
using  times  as  abscissae  and  temperatures  as  ordinates,  mak- 
ing the  scale  as  large  as  possible.     Discuss  the  form  of  the 
curves  and  the  relation  between  the  several  curves.  What  re- 
lation exists  between  absorption  and  radiation  at  the  highest 
and  at  the  lowest  temperatures     reached?     Connect  the  re- 
sults with  the  fact  that  stoves  are  made  black  and  the  fender 
and  knobs  of  the  stove  are  nickeled. 

53.     RATIO  OF    THE    TWO    SPECIFIC    HEATS    OF 

AIR. 

Reference. — Duff,  p.  264. 

The  object  of  this  experiment  is  to  obtain  the  value  of  the 
ratio  y  of  the  specific  heat  of  air  at  constant  pressure  to  it* 
specific  heat  at  constant  volume.  The  method  employed  is  a 
modification  of  that  used  first  by  Clement  'and  Desormes.  A. 
quantity  of  the  gas,  compressed  in  a  large  flask,  is  momentarily 


114  RATIO  OF  THE  TWO  SPECIFIC  HEATS  OF  AIR.  [53 

put  in  communication  with  the  atmosphere  to  allow  its  pres- 
sure to  fall  adiabatically  to  atmospheric  pressure,  its  temper- 
ature simultaneously  falling  a  little.  The  gas,  when  shut  off 
again  from  the  atmosphere,  gradually  warms  up  to  its  initial 
temperature,  causing  an  appreciable  rise  in  its  pressure.  Let 
/>,be  the  pressure  in  the  compressed  gas  at  the  start,  v±  the 
volume  of  unit  mass  of  the  gas  and  f±  its  temperature  (the 
same  as  that  of  the  room).  Let  pQ,  v2,  and  tz  be  the  corres- 
ing  values  of  these  quantities  immediately  after  communica- 
tion between  the  compressed  gas  and  the  atmosphere  is  es- 
tablished. Then  p2,  v2,  and  £x  will  be  the  values  of  these 
same  quantities  at  the  end^  if  p2  is  the  final  pressure.  The  gas 
has  now  been  in  three  conditions,  as  follows : 
Condition  Pressure  Vol.  of  igm.  Temperature 

I.  p,  v,  t, 

II.  p0  v2  t2 

III.  p2  v2  t, 

The  change  from  I  to  II  was  adiabatic,  since  no  time  was  al- 
lowed for  heat  to  pass  in  or  out  of  the  gas  by  conduction  or  ra- 
diation ;  hence,  by  the  law  for  adiabatic  changes  in  a  perfect 
gas, 

(0          s^A^z'/A.   or  O2/"i/=A/A- 
The  change,  from  I  to  III  was  isothermal;  hence,  by  Boyle's 
law, 

(2)  "ViA  =  ^A»    or     (v^/Vif  =  (A /A/ 

Hence    (pl  /  p.,  /  :=  (pl  I  pQ  );    or,  taking  the  logarithm  and 
solving  for  Y 

(^  log  A -log  A 

logA-logA' 

The  desired  ratio  may  be  obtained,  experimentally,  there- 
fore, by  observing  the  values  of  the  three  pressures. 

The  apparatus  consists  of  a  large  carboy  provided  with  a 


53  RATIO  OF  THE:  TWO  SPECIFIC  HEATS  OF  AIR.  115 

large-bore  stop-cock  so  that  the  enclosed  space  may  at  pleas- 
ure be  opened  to,  or  shut  off  from,  the  atmosphere.  The  pres- 
sure of  the  enclosed  air  is  measured  by  an  oil  manometer, 
whilst  air  can  be  forced  into  or  withdrawn  through  another 
inlet.  To  thoroughly  dry  the  enclosed  air,  some  strong  sul- 
phuric acid  is  poured  into  the  bottom  of  the  carboy. 

(a)  Close  the  stop-cock,  and  with  a  bicycle  pump  intro- 
duce enough  air  in  the  carboy  to  give  a  reasonably  large  dif- 
ference of  pressure,  as  indicated  by  the  manometer.     Shut  off 
connection  between  the  carboy  and  the  pump,  and  wait  a  few 
minutes  until  the  temperature  of  the  enclosed  air  is  the  same 
as  that  of  the  room,  which  will  be  when  the  manometer  shows 
a  steady  ?  constant  pressure.    Read  the  manometer  and  the  bar- 
ometer. To  get  the  value  of  the  pressure-difference  recorded  by 
the  manometer,  it  will  be  necessary  to  know  the  density  of  the 
oil  used.    This  is  posted  on  the  apparatus. 

(b)  Open  the   stop-cock   wide  and   thus   connect   the  en- 
closed air  with  the  atmosphere.    Leave  open  only  for  a  second, 
then  close  again.     Wait  some  time  until  the  temperature  of 
the  enclosed  air  has  risen  again  to  that  of  the  room,  as  indi- 
cated by  a  steady,  constant  difference  in  pressure ;  then  read  the 
manometer. 

(c)  Using  the  data  in  (a)  and  (b),  determine  from  equa- 
tion (3)  the  value  of  y  for  air. 

(d)  Repeat  (a),  (&),  and  (c)  two  or  three  times,  and  take 
the  mean  of  the  results. 

Obtain  from  the  Tables  the  values  of  the  two  specific  heats 
of  air,  calculate  their  ratio,  and  compare  with  the  result  just 
found  by  experiment. 

Point  out  the  principal  sources  of  error,  stating  how  each 
affects  the  result. 

Explain  why  the  specific  heat  of  a  gas  at  constant  pressure 
should  be  greater  than  its  specific  heat  at  constant  volume. 


n6 


LOGARITHMS. 


0 

1 

2 

3 

4 

.5 

6 

7 

8 

9 

123 

456 

789 

10 

ITT 

12 
1  13 
114 
15 
|161 

0000 

0043 

0086 

0492 
0864 
1206 

0128 

0170 

O2I2 

0253 

0294 

0334 

0374 

Use  Table  on  p.  118. 

0414 

0792 
1139 

0453 

0828 

H73 

0531 
0899 
1239 

0569 
0934 
1271 

0607 
0969 
1303 

0645 
100^ 
1335 

0682 
1038 
1367 

6719 
1072 
1399 

0755 
iro6 
T430 

4811 
3710 
3  6  10 

15  19  23 
14  17  ax 

13  16  19 

26-30  34 
24  28  31 
23  26  29 

F46l 
I76l 
2041 

1492 
1790 
2068 

1523 
1818 
2095 

1553 
1847 

2122 

1584 
1875 
2148 

1614 
1903 
2175 

164^ 
!93i 

2201 

1673 

1959 
2227 

1703 
1967 
2253 

1732 
2014 
2279 

36  9 
36  8 
35  8 

12  15  18 
ix  14  17 
ii  13  16 

21    24   27 
20  22    25 

18  21  24 

17 
18 
19 

2304 

2553 
2788 

2330 
2577 
2810 

2355 
2601 
2833 

2380 
2625 
2856 

2405 
2648 

2878 

2430 
2672 
29OO 

2455 
2695 
2923 

2480 
2718 
2945 

2504 
2742 
2967 

2529 
2765 
2989 

5   7 
5  7 
4   7 

IO    12    15 

9  '«  *4 

9  IX  *3 

17    20   22 

16  19  21 

16  18  20 

20 
|2T 
22 
23 
124 
25 
126 

3010 

3032 

3054 

3075 

3096 

3H8 

3139 

3160 

3181 

3201 

4  6 

8  ii  13 

15  17  19 

3222 

3424 
3617 

3243 
3444 
3636 

3263 
3464 
3655 

3284 
3483 
3674 

3304 
3502 
3692 

3324 
3522 
3711 

3345 
3541 
3729 

3365 
3560 
3747 

3385 
3579 
3766 

3404 
3598 
3784 

4  6 
4  6 
4  6 

8    10    12 
8    10   12 

7    9  11 

14  16  18 
14  15  i-7 
»3  '5  '7 

3802 

3979 
4^0 

3820 

3997 
4166 

3838 
4014 
4183 

3856 
4031 
4200 

3874 
4048 
4216 

3892 
4065 
4232 

39<>9 
4082 

4249 

3927 
4099 
4265 

3945 
4116 
4281 

3962 
4133 
4298 

4   5 
3   5 

3   5 

7    9  ii 
7    9  10 
7    8  10 

12  14  16 

12-  14    15 

ix  13  15 

27 
28 
29 

43M 
4472 
4624 

4330 
^87 
4639 

4346 
4502 
4654 

4362 
45I8 
4669 

4378 
4533 
4683 

4393 
4548 
4698 

4409 
4564 
4713 

4425 
4579 
4728 

4440 
4594 
4742 

4456 
4609 
4757 

3   5 
3   5 
3  4 

689 
6    8    9 
679 

II    13    I4 
IX    12    14 
IO    12    13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

3   4 

6    7    9 

10  ii   13 

31 
132 
133 

4914 
5051 

5185 

4928 
5065 
5198 

4942 
5079 
5211 

4955 
5092 
5224 

4969 
5105 
5237 

4983 
5"9 
5250 

4997 
5132 
5263 

5011 

5M5 
5276 

5024 
5159 
5289 

5038 
5172 
5302 

3  4 
3  4 
3  4 

6    7    8 
5    7    8 
5    6    8 

10    II    12 
9    11    12 

9  10  12 

134 
35 
36 
137 
38 
39 

5315 
5441 
5563 

5328 
5453 
3575 

5340 
5465 
5587 

5353 
5478 
5599 

5366 
5490 
5611 

5378 
5502 
5623 

5391 
5514 
5635 

5403 
5527 
5647 

54i6 

5539 
5658 

5428 

5551 
5670 

3  4 

4 
4 

5    6    8 
5    6    7 
5    6    7 

9  10  ii 
9_io  ii 
8  10  ii 

5682 
5798 
59" 

5694 

S8°9 
5922 

5705 
5821 

5933 

5717 
5832 
5944 

5729 
5843 
5955 

5740 
5855 
5966 

5752 
5866 
5977 

5763 
5877 
5988 

5775 
5888 

5999 

5786 

5899 
6010 

3 
3 
3 

5    6    7 
5    6    7 
457 

8    9  10 
8    9  to 
8    9  10 

40 
41 
42 
43 
44 
45 
46 

to 

48 

49 
[50 

6021 

6031 

6042 

605.3 

6064 

6075 

6085 

6096 

6107 

6117 

3 

4     5    6 

8    9  10 

6128 
6232 
6335 

6138 
6243 
6345 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
6274 
6375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 
6314 
6415 

6222 
6325 
6425 

3 

3 
3 

4     S    6 
4     5    6 
4     5    6 

7     8    9 
7     8    9 
7    8    Q 

6435 
6532 
6628 

6444 

6542 
5637 

6454 
6551 
6646 

6464 
6561 
6656 

^474 
6571 
6665 

6484 
6580 
6675 

6493 
6590 
6684 

6503 
6599 
6693 

6513 
6609 
6702 

6522 
6618 
6712 

3 
3 

4     5     6 
4    5    6 

7     8    9 

7     8    9 

6721 
6812 
6902 

6730 
6821 
6911 

6739 
6830 
6920 

6749 

6839 
6928 

6758 
6848 
6937 

6767 
6857 
6946 

6776 
6866 
6955 

6785 
6875 
6964 

6794 
6884 
6972 

6803 
6893 
6981 

3 
3 
3 

455 
445 
445 

6    7    8 
6    7    fa 
6    7    8 

6990 

,998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

3 

345 

6    7     8 

51 
52 
53 

54 

7076 
7160 
7243 

7084 
168 

251 

7093 
7177 
7259 

7101 

7185 
7267 

7110 
7193 

7275 

7118 
7202 
7284 

7126 
7210 
7292 

7135 
7218 
7300 

7143 
7226 
7308 

7152 
7235 
73i6 

3 

2 
2 

345 
345 
345 

6    7     8 
6    7     7 
667 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

738o 

7388 

7396 

I    2    2 

345 

6    6    7 

LOGARITHMS. 


117 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

123 

456 

789 

55 

740-1 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

I  2  2 

345 

5  6  7 

56 
57 
58 

7482 
7559 
7634 

7490 
7566 
7642 

7497 
7574 
7649 

7505 
7582 
7657 

7513 
7589 
7664 

7520 

7597 
7672 

7528 
7604 
7679 

7536 
7612 
7686 

7543 
7619 
7694 

755J 
7627 
7701 

I  2  2 
I  2  2 
f  I  2 

345 
3  4 
3  4 

567 
5  6  7 
5  6  7 

59 
60 
61 

7709 

7732 
78S3 

7716 
7789 
7860 

7723 
7796 
7868 

773i 
7803 
7875 

7738. 
7810 

7882 

7745 
7818 
7889 

7752 
7825 
7896 

7760 
7832 
79<>3 

7767 

7839 
7910 

7774 
7846 

7917 

112 
112 
I  1  2 

3  4 
3  4 
3  4 

5  6  7 

5  6  t 
5  6  6 

62 

63 
64 

7924 

7993 
8062 

7931 
Sooo 
8069 

7938 
8007 
8075 

7945 
8014 

8082 

7952 
8021 
8089 

7959 
8028 
8096 

7966 
8035 
8102 

7973 
8041 
8109 

7980 
8048 
8116 

7987 
8055 
812? 

I  I  2 
112 
I  I  2 

112 

3  3 
3  3 
3  3 

566 
5  5  6 
5  5  6 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

3  3 

5  5  t 

66 
67 
68 
69 
70 
71 

8195 
8261 
8325 
8388 
8451 
8513 

8202 
8267 
8331 
8395 
8457 
8519 

8209 

8274 
8338 

8215 
8280 
8344 

8222 
8287 
8351 

8228 
8293 
8357 

8235 
8299 
8363 

8241 
8306 
8370 

8248 
8312 
8376 

8254 

8319 
8382 

8445 
8506 

8567 

112 
I  I  2 
I  I  2 

33    556 
33    556 
33    456 

8401 
8463 
8525 

8407 
8470 
853i 

8414 
8476 
8537 

8420 
8482 
8543 

8426 
8488 
8549 

8432 
8494 

8555 

8439 
8500 
8561 

112 

2  3 

455 

72 
73 
74 

8573 
8633 
8692 

875i 

8579 
8639 
8698 

8585 
8645 
8704 

8591 
8651 
8710 

8597 
8657 
8716 

8603 
8663 
8722 

8609 
8669 
8727 

8615 
8675 
8733 

8621 
8681 
8739 

8627 
8686 
8745 

I  I  2 

ri2 

112 

112 

2  3 
2  3 
2  3 

455 

5  5 
5  5 

75 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

233 

'   5  ^ 

4 
4 

76 
77 
78 

8808 
8865 
8921 

8814 
8871 
8927 

8820 
8876 
8932 

8825 
8882 
8938 

8831 
8887 
8943 

8837 
8893 
8949 

8842 
8899 
8954 

8848 
8904 
8960 

8854 
8910 
8965 

885-9 
8915 
8971 

112 
I  I  2 
I  I  2 

233 
233 
233 

79 
80 
81 

8976 
9031 
9085 

8982 
9036 
0090 

8987 
9042 
9096 

8993 
9047 
9101 

8998 

9053 
9106 

9004 
9058 
9112 

9009 
9063 
9117 

9015 
9069 
9122 

9020 
9074 
9128 

9025 
9079 
9133 

112 
1  I  2 
I  I  2 

233 
233 
233 

4 
4 
4  4 

82 
83 
84 

9138 
9191 
9243 

9143 
9196 
9248 

9149 
9201 
9253 

9154 
9206 
9258 

9*59 
9212 
9263 

9165 
9217 
9269 

9170 
9222 
9274 

9iZ5 
9227 
9279 

9180 
9232 
9284 

9186 
9238 
9289 

112 
112 
I  I  2 

233 
233 
233 

4  4 

:: 

85 
"86 
87 
88 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

233 

4  4  5 

9345 
9395 
9445 

9350 
9400 
9450 

9355 
9405 
9455 

9360 
9410 
9460 

9365 
9415 
9465 

9370 
9420 
9469 

95i8 
9566 
9614 

9375 
9425 
9474 

938o 
9430 
9479 

9385 
9435 
9484 

939° 
9440 
9489 

0  I  1 
0  I  I 

2     A 
2      3 

3 

3 

89 
90 
91 

9494 
9542 
959^ 

9499 
9547 
9595 

9504 
9552 
9600 

9509 
9557 
9605 

9513 
9562 
9609 

9523 
957i 
9619 

9528 
9576 
9624 

9533 
958i 
9628 

9538 
9586 
9633 

Oil 
Oil 
Oil 

2     3 
2    3 
2     3 

3 
3 

3 

92 
93 
94 

9638 
9685 
9731 

9643 
9689 
9736 

9647 
9694 
974i 
9~7S6 

9652 
9699 
9745 

9657 
9703 
9750 

9661 
9708 
9754 

9666 
97*3 
9759 

9671 
9717 
9763 

9675 
9722 
9768 

9680 
9727 
9773 

Oil 
Oil 
0  I  I 

2     3 
2    3 
2  -   3 

3 
3 
3  4 

95 

9777 

9782 

979> 

9795 

9800 

9805 

9809 

9814 

9818 

0  1  I 

3     3 

3  4 

96 
97 
98 

9823 
9868 
9912 

9827 
9872 
9917 

9832 

9877 
9921 

9836 
9881 
9926 

9841 
9886 
9930 

9845 
9890 

9934 

9850 
9894 
9939 

9854 
9899 

9943 

9859 
99°3 
9948 

9863 
9908 
9952 

0  I  I 
0  I  I 
0  I  I 

2    3 

2      3 
2      3 

344 
344 
344 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

999  1 

9996 

0  I  I 

223 

334 

LOGARITHMS. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

ooooo 

043 

087 

130 

173 

217 

260 

303 

346 

389 

101 
102 
103 

432 
860 
oi  284 

475 
903 
326 

518 
945 
368 

56i 
988 
410 

604 
030 

452 

647 
072 

494 

689 
"5 
536 

732 
157 

578 

775 
199 
620 

817 
242 
662 

104 
105 
106 

703 

02  119 

531 

745 
160 

572 

787 

202 

612 

828 
243 
653 

870 
284 
694 

912 
325 

735 

953 
366 
776 

995 
407 
816 

036 
449 

857 

078 
490 
898 

107 
108 
109 

938 
03342 

743 

979 

383 
782 

OI9 

423 
822 

060 

463 
862 

100 

503 
902 

141 

543 
941 

181 

583 
981 

222 
623 
O2  1 

262 
663 
060 

302 

703 

100 

INDEX. 


Absolute,     expansion     of     mer- 
cury,  55. 
humidity,  97. 

Acceleration,  average  linear,  25. 
uniform,  25. 
due  to  gravity,  31. 
machines,  25,  26. 
normal,  27. 
Air-buoyancy,  correction  for,  in 

weighing,  3. 
Air,  density  of,  21, 
thermometer, 
constant  pressure,  89. 
constant  volume,  91. 
Apparent  expansion,  of  alcohol, 

62. 

Archimedes'  principle,  3,  53. 
Auguste's  psychrometer,  98. 
Balance,  Jolly's,  4. 
model,   7. 
sensitive,   I. 

Barodeik,  calibration  of  the,  99. 
Boyle's  law,  experimental  study 

of,  ii. 

Brake,  Prony,  no. 
Callendar's      method     for      me- 
chanical   equivalent    of    heat, 
69. 
Capillarity,    rise    of    liquids    in 

tubes,  37. 

Centripetal  force,  28. 
Charles'  Law,  22;  tested,  89,  91. 
Coefficient      of      expansion,      of 
liquid,   by      Archimides'    prin- 
ciple,  52. 

by  Regnault's  method,  55. 
by  pycnometer,  61. 
of    glass     by    weight -ther- 
mometer, 58. 
Coefficient,  of  friction,  102. 

of  restitution,  104. 
of  rigidity,   108. 


Coefficient  of  expansion,   of  air 
at     constant     pressure;     flask 
method,  86. 
with  air  thermometer,  89. 

Coincidences,  method  of,  32. 

Component  of  force,  38. 

Composition    and    resolution    of 
forces,    15,    17. 

Constant  -  pressure     air -thermom- 
eter, 89. 

Constant-volume      air-thermom- 

eter,  91. 

Contact,  angle  of,  38. 
Cooling  curve,  74. 
Cooling,  law  of,  67. 
Correction    of   a    mercury   ther- 
mometer, 46. 
Dalton's  Law,  97. 
Density,    of    air,    determination 

of,  21. 

of  carbon  dioxide,  de- 
termination of,  23. 
of  a  cylindrical   solid, 

3- 
of  solids,  with  Jolly's 

balance,  4. 
Dew-point,  97. 

hygrometer,  98. 
Double  weighing,  4. 
Equation,  the  force,  34. 

of  moments,  30. 
Expansion,   apparent   and   abso- 
lute,  62,   64. 
absolute,     of     olive 

oil,    55- 
Force,       centripetal,       definition 

of,   27. 
formula,   28. 
equation,  the,  34. 
table,  vector  sum  using 

the,   15. 

Forces,    experimental    study    of 
three,   17. 


120 


INDEX 


Freezing  point,  of  solutions,  81. 
Friction,  coefficient  of,   102. 
"G"   determination   of,  with  the 
fall-machine,  24;   with   pendu- 
lum, 31. 
Glass,    coefficient    of    expansion 

of,  58. 

Gram-molecular  weight,  85. 
Heat,  capacity,  65. 

of   fusion,    Wood's    Alloy, 

76. 
mechanical  equivalent  of, 

69. 

of   neutralization,   85. 
of   solution,   83. 
of     vaporization,     at     the 
boiling  point,  78. 
at   room   temperature,  80. 
Hooke's  Law,  106,  108. 
Humidity,  absolute,  97. 
relative,  97. 
Hygrometer,      Regnault's      wet- 

and-dry  bulb,  97. 
Impact,  elastic,  104. 

inelastic,    105. 
Jolly's  balance,  4. 
Kinetic    energy,    loss    of    in    im- 
pact,   104. 

Lever  arm,  defined,  29. 
Liquid,  density  of,  6. 
Logarithms,     table    of,    4-place, 

1 16. 

Mechanical    equivalent    of    heat, 
by   Callendar's  method,  69. 
by  Puluj's  method,  71. 
Method  of  coincidences,  32. 
of  cooling,  67. 
of  mixtures,  76. 
of  heating,  66. 
of   vibrations,    I. 
Modulus,  of  torsion,  108. 

Young's,    by    stretch- 
ing,   1 06. 

Moments,  principle  of,  30. 
Momentum,      conservation      of, 

103. 

Motion,  study  of,  uniformly  ac- 
celerated, 24. 


Normal,  acceleration,  28. 

force,  27. 

Normal  solution,  85. 
Parallelogram  of  forces,  16. 
Pendulum,  the  simple,  31. 
Points,    fixed,    of    thermometer, 

4.6. 

Principle  of  moments,  29. 
Prony  brake,   no. 
Pycnometer,     expansion     of     a 

liquid  by,  61. 
Radiation,  rate  of,   112. 
Ratio  of  specific  heats,  114. 
Relative  humidity,  97. 
Resolution  of  forces,   15,  17. 
Restitution,  coefficient  of,  104. 
Resultant,  of  two  forces,   16. 
Rider,  use   of  the  centigram,   2. 
Rigidity,  coefficient  of,   108. 
Sensitiveness,     of    balance,     de- 
fined, 2. 
formula  for,  8. 
Solution,  heat  of,  83. 
normal,   85. 

Solutions,  freezing  point  of,  81. 
Specific  heat  of  a  liquid 
by  method  of  heating,  66. 
by  method  of  cooling,  67. 
Surface  tension,  by  Jolly's  bal- 
ance, 36. 

in  capillary  tubes,  37. 
between  plates,  39. 
Tension,      surface,      by      direct 

measurement,  36. 
Thermometer,   absolute   calibra- 
tion of,  45. 
comparison      of      alcohol 

and  water,  54. 
constant-  pressure   air-,  89. 
relative  calibration  of,  49. 
Torsion,  modulus  of,  108. 
Twisting,  Hooke's  law  for,   108. 
Uniform,      accelerated     motion, 

24. 
circular    motion,    26. 

Vapor  pressure,  and  volume,  93. 


INDEX 


121 


Variation,    of    boiling    point    of 

water  with  pressure,  50. 
Vibrations,   method   of,  2. 
Viscosity,  coefficient  of,  42. 
Volumenometer,  the,   13. 
Water,  equivalent  of  a  body,  65. 
equivalent     of     a     ther- 
mometer,  65. 
expansion    curve    of,   63. 


Weighing,  by  method  of  vibra- 
tions,   i. 

method  of  double,  4. 
Weight  thermometer,  58. 
Weights,  molecular,  82. 
Wet-and-dry,     bulb     thermome- 
ter, 97. 

Young's  modulus,  by  stretching, 
1 06. 


MEASUREMENT   OF   PHYSICAL 
QUANTITIES. 


Experimental  work  has  one  of  two  objects ;  either  to  find  out 
what  kind  of  a  result  follows  under  given  conditions,  or  to  find 
out  the  numerical  relations  between  different  quantities.  The 
first  class  of  experiments  is  called  qualitative,  the  second  quan- 
titative. In  the  earlier  days  of  any  science  qualitative  experi- 
ments are  numerous ;  when  the  science  is  more  mature,  the  ma- 
jority of  the  experiments  are  quantitative.  The  determination 
of  various  quantitative  relations  is  the  object  of  physical  meas- 
urement. 

In  making  a  physical  measurement,  the  magnitude  of  each 
quantity  concerned  has  to  be  expressed  in  terms  of  some  unit, 
and  the  process  of  measurement  consists  essentially  in  finding 
how  many  times  this  unit  is  contained  in  the  given  quantity. 
The  distance  between  two  points,  for  example,  may  be  ex- 
pressed in  terms  of  the  number  of  foot  rules  which  could  be 
laid  end  to  end  between  these  points. 

Some  quantities  can  thus  be  measured  directly,  but  others  can 
be  measured  only  indirectly.  Thus  the  density  of  a  solid  cylin- 
der of  any  substance  cannot  be  experimentally  determined  by 
finding  how  many  times  the  unit  of  density  is  contained  in  the 
density  of  the  cylinder.  It  would  be  determined  usually  by 
measuring  the  mass,  length,  and  diameter  of  the  cylinder,  and 
from  them  calculating  the  density.  The  great  majority  of 
physical  measurements  are  indirect  measurements. 

ERRORS. 

Every  measurement  is  subject  to  errors.  In  the  simple  case 
of  measuring  the  distance  between  two  points  by  means  of  a 
meter  rod,  a  number  of  measurements  usually  give  different 


124  MEASUREMENT  Ol-   PHYSICAL  QUANTITIES. 

results,  especially  if  the  distance  is  several  meters  long  and 
the  measurements  are  made  to  small  fractions  of  a  millimeter. 
The  errors  introduced  are  due  in  part  to — 

(1)  Inaccuracy  of  setting  at  the  starting  point. 

(2)  Inaccuracy  of  setting  at  intermediate  points  when  the 
distance  exceeds  one  meter. 

(3)  Inaccuracy  in  estimating  the  fraction  of  a  division  at 
the  end  point. 

(4)  Parallax  in  reading;  that  is,  the  line  from  the  eye  to 
the  division  line  read  is  not  perpendicular  to  the  scale,  and, 
where  both  eyes  are  used,  the  imaginary  line  joining  the  two 
eyes  is  not  parallel  to  these  division  lines. 

(5)  The  meter  rod  not  being  straight. 

(6)  The  temperature  not  being  that  for  which  the  meter 
red  was  graduated. 

(7)  Irregular  spacing  of  divisions. 

(8)  Errors  in  the  standard  from  which  the  division  of  the 
meter  rod  was  copied. 

Besides  the  above  there  are  doubtless  other  sources  of  error. 
It  may  be  well  here  to  note  that  blunders,  such  as  mistakes 
due  to  mental  confusion  in  putting  down  a  wrong  reading, 
or  mistakes  in  making  an  addition,  are  not  usually  classed 
as  errors. 

Of  the  above  errors,  (i),  (2),  and  (3)  can  be  very  much 
decreased  by  having  fine  divisions  on  the  scale  and  by  reading 
with  microscopes ;  (4)  can  be  made  small  by  bringing  the  scale 
on  the  meter  rod  close  to  the  object  to  be  measured;  (5)  can 
be  made  very  small  by  using  a  meter  rod  of  special  design,  or, 
in  rough  work,  by  holding  the  meter  rod  against  a  straight 
edge;  (6)  can  be  nearly  eliminated  by  using  the  meter  rod  only 
at  the  proper  temperature,  or,  if  its  temperature  and  co- 
efficient of  expansion  are  known,  by  calculating  a  correction 
to  be  applied;  (7)  can  be  diminished  only  by  a  careful  com- 
parison of  the  lengths  of  the  different  divisions;  and  (8)  can 
only  be  corrected  for  when  something  is  known  of  the  relative 


MEASUREMENT  OF  PHYSICAL  QUANTITIES.  125 

accuracy  of  the  standard  from  which  the  meter  rod  was  copied. 
But  even  with  the  most  refined  methods  and  the  most  care- 
ful application  of  corrections,  different  measurements  of  the 
same  distance  usually  give  different  results. 

Errors  due  to  (6),  (7),  and  (8)  may  be  determinate  errors, 
that  is.,  errors  for  which  more  or  less  accurate  corrections  can 
be  calculated;  whereas  those  due  to  (i),  (2),  and  (3)  are 
indeterminate  errors,  that  is,  errors  for  which  corrections  can- 
not be  calculated.  Moreover,  of  those  errors  for  which  cor- 
rections are  not  applied,  some,  like  those  due  to  (i),  (2),  and 
(3),  will  be  variable  in  amount  and  will  tend  to  make  the 
value  obtained  sometimes  too  large  and  sometimes  too  small; 
while  others,  like  those  due  to  (7)  and  (8),  when  corrections 
for  them  are  not  applied,  will  be  constant  and  will  tend  to  make 
the  value  obtained  always  too  large  or  always  too  small. 

Since  the  average  value  of  those  variable  errors  which  tend  to 
make  a  result  too  large  will  after  a  considerable  number  of 
measurements  be  about  the  same  as  the  average  value  of  those 
variable  errors  which  tend  to  make  the  result  too  small,  the 
mean  of  a  large  number  of  measurements  is  usually  nearly 
free  from  variable  errors.  In  order  as  nearly  as  possible  to 
do  away  with  constant  errors,  the  same  quantity  should  be  meas- 
ured by  as  many  different  methods  as  possible.  The  results  by 
the  different  methods  will  usually  differ  somewhat,  but  from 
them  all  a  value  can  be  calculated  which  is  probably  nearer  the 
true  value  than  is  any  one  of  the  separate  results. 

TRUSTWORTHY  AND  SIGNIFICANT  FIGURES. 

Since  all  measurements  are  subject  to  errors,  it  is  important 
to  be  able  to  determine  how  many  figures  of  a  result  can  be 
trusted. 

In  direct  measurements  it  is  usually  possible  to  make  a  fairly 
accurate  estimate  of  the  extent  to  which  a  reading  can  be 
trusted.  Thus  in  reading,  by  means  of  the  unaided  eye,  the 
position  of  a  fine  line  which  crosses  a  meter  rod,  the  reading 


126  MEASUREMENT  OF  PHYSICAL  QUANTITIES. 

will  not  be  in  error  by  so  much  as  a  millimeter,  but  pretty 
surely  will  be  in  error  by  more  than  a  thousandth  of  a  milli- 
meter. So  the  extent  to  which  the  reading  can  be  trusted 
will  lie  between  these  limits.  A  person  who  is  accustomed  to 
estimating  fractions  of  a  small  division  will  be  rather  sure  of 
not  making  an  error  so  great  as  the  tenth  of  a  millimeter,  and 
he  can  often  trust  his  reading  to  a  twentieth  of  a  millimeter. 

It  is  convenient  always  to  put  down  all  the  figures  that  can 
be  trusted,  even  if  some  of  them  are  ciphers.  Thus  the  state- 
ment that  a  distance  is  50  cm.  implies  that  there  is  reason  for 
supposing  that  the  distance  really  lies  between  49  cm.  and  51 
cm.,  whereas  the  statement  that  the  distance  is  50.00  cm.  im- 
plies that  there  is  reason  for  supposing  that  the  distance  really 
lies  between  49.99  cm.  and  50.01  cm.  When  the  distance  is 
said  to  be  50  cm.,  the  second  figure  is  the  last  in  which  any 
confidence  can  be  placed ;  when  the  distance  is  said  to  be  50.00 
cm.,  the  fourth  figure  is  the  last  in  which  any  confidence  can 
be  placed.  This  fact  is  indicated  by  saying  that  in  the  first 
case  the  quantity  can  be  trusted  to  the  second  significant  figure, 
and  in  the  other  case  to  the  fourth.  By  the  number  of  signifi- 
cant figures  in  a  quantity  is  meant  the  number  of  trustworthy 
figures,  counting  from  the  left,  irrespective  of  the  decimal 
point;  thus  there  are  two  significant  figures  in  0.000026.  If 
ei  distance  is  about  50000  km.  and  the  third  significant  figure 
is  the  last  in  which  any  confidence  can  be  placed,  this  fact  may 
be  indicated  by  saying  that  the  distance  is  50.0  X  io3  km. 

In  indirect  measurements  the  result  is  usually  calculated  by 
some  formula.  To  find  out  how  many  figures  should  be  kept 
in  the  result  consider  the  following  two  cases : — 

I.  If  the  result  is  the  algebraic  sum  of  several  quantities, 
such  as  214.3,  3641,  and  0.506,  it  is  seen  that  in  the  sum 
251.216,  no  figure  beyond  that  in  the  first  decimal  place  can 
be  trusted,  because,  in  the  quantity  which  has  the  fewest  trust- 
worthy decimal  places,  namely  214.3,  no  figure  beyond  the  3 
can  be  trusted,  otherwise  it  would  have  been  expressed.  So 


MEASUREMENT  OF  PHYSICAL  QUANTITIES.  127 

the  sum  will  not  be  written  251,216,  but  will  be  written  251.2. 
This  suggests  the  following  rule  : — 

RULE  L  In  sinus  and  differences  no  more  decimal  places 
should  be  retained  than  can  be  trusted  in  the  quantity  having 
fewest  trustworthy  decimal  places. 

2.  If  the  result  is  the  product  of  two  quantities,  such  as 
314.428  and  32.6,  then  the  product  cannot  be  trusted  to  more 
significant  figures  than  appear  in  the  quantity  having  fewest 
trustworthy  figures,  irrespective  of  the  decimal  place.  To  make 
this  clear,  consider  the  following  products  : — 

314.428  X  32.5  =  10218.9100 

314.428  X  32.6  =  10250.3528 

314.428  X  327  —  10281.7956 

314.        X  32.6  =  10236.4 

It  is  seen  from  the  first,  second  and  third  products  that  if 
the  quantity  which  is  supposed  to  be  32.6  is  really  32.5  or  32.7, 
then  after  the  first  three  significant  figures  the  true  value  of  the 
product  differs  materially  from  the  value  obtained.  The  sec- 
ond and  fourth  of  the  above  products  show  that  if  more  than 
three  significant  figures  cannot  be  trusted  in  one  of  two  quan- 
tities which  are  to  be  multiplied,  it  is  not  worth  while  to  use 
more  than  three,  or  at  most  four,  significant  figures  of  the  other. 
The  product  in  this  case  would  be  written  1.02  X  io4,  or  at 
most  1.024  X  io4.  These  facts  suggest  the  following  rule: 

RULE  IL  In  products  and  quotients  no  more  significant 
figures  should  be  kept  than  can  be  trusted  in  the  quantity  having 
fewest  trustworthy  figures. 

Until  the  final  result  is  reached,  it  is  often  worth  while  to 
keep  one  more  figure  than  the  above  rules  indicate. 
For  logarithms  a  safe  rule  is  the  following : — 
RULE  III.     When  any  of  the  quantities  which  are  to  be  mul- 
tiplied or  divided  can  be  trusted  no  closer  than  o.oi  of  one  per 
cent.,  use  a  five-place  table;  when  any  of  them  can  be  trusted 


128  MEASUREMENT  OF  PHYSICAL  QUANTITIES. 

no  closer  than  o.i  of  one  per  cent.,  use  a  four-place  table;  and 
when  any  of  them  can  be  trusted  no  closer  than  i  per  cent., 
use  a  slide  rule. 

REQUIRED  ACCURACY  OF  MEASUREMENT. 

From  Rule  I  it  will  be  seen  that  if  a  small  quantity  is  to  be 
added  to  a  large  one,  the  percentage  accuracy  of  the  measure- 
ment of  the  small  quantity  need  not  be  so  great  as  that  of  the 
large  one.  Thus  if  H  =  a  +  b,  and  if  a  is  about  100  cm.  and 
b  about  i  cm.,  a  I  per  cent,  error  in  a  will  produce  in  H  no 
greater  effect  than  a  100  per  cent,  error  in  b.  When  quantities 
are  to  be  added  or  subtracted,  they  should  be  measured  to  the 
same  number  of  decimal  places. 

From  Rule  II  it  will  be  seen  that  if  a  small  quantity  and  a 
large  one  are  to  be  multiplied,  the  percentage  accuracy  of  the 
measurement  of  the  small  quantity  should  be  at  least  as  great 
as  that  of  the  large  one.  Thus  if  H  =  ab,  a  i  per  cent,  error  in 
a  will  produce  in  H  the  same  effect  as  a  i  per  cent,  error  in  b. 
So  that  if  a  is  about  100  cm.,  and  b  is  about  i  cm.,  and  if  b 
cannot  be  trusted  closer  than  o.oi  cm.,  there  is  no  gain  in  ac- 
curacy by  measuring  a  much  closer  than  i  cm.  When  quanti- 
ties are  to  be  multiplied  or  divided  they  should  be  measured  to 
within  the  same  fractional  part  of  themselves,  for  example, 
all  of  them  within  i  per  cent,  and  none  of  them  much  closer,  or 
all  of  them  within  o.oi  of  one  per  cent  and  none  of  them  much 
closer. 

The  last  statement  needs  modification  in  the  case  of  a  power. 
If  the  value  found  for  a  quantity  a  is  i  per  cent  too  large,  that 
is,  is  i .010,  then  the  value  that  will  be  obtained  for  a~  is 
[.020 1  a,  which  is  about  2  per  cent  too  large,  and  the  value  ob- 
tained for  a3  is  1.0303010,  which  is  about  3  per  cent,  too  large. 
In  general,  if  the  value  found  for  a  is  k  per  cent  too  large,  the 
value  that  will  be  obtained  for  on  will  be  nk  per  cent,  too  large. 
So  that  a  quantity  which  is  to  be  squared,  cubed,  or  raised  to 


MEASUREMENT  OF  PHYSICAL  QUANTITIES.  129 

some  higher  power  should  be  measured  with  more  care  than  if 
it  entered  the  formula  only  to  the  first  power. 

It  is  evident,  then,  that  a  preliminary  study  of  the  required 
accuracy  of  measurement  will  not  only  save  much  time,  by 
pointing  out  those  quantities  which  need  to  be  measured  with 
only  a  rough  accuracy,  but  will  also  serve  to  determine  those 
quantities,  usually  the  smallest,  in  the  measurement  of  which 
great  care  must  be  taken  and  sensitive  instruments  used. 

(Largely  reproduced  from  Perry  &  Jones.) 


I3O  MEASUREMENT  OF  PHYSICAL  QUANTITIES. 

TABLE  I. 


USEFUL  NUMERICAL  RELATIONS. 

Mensuration. 

Circle:     circumference  =  2^r\  area  = 
Sphere:     area  =  4^r;  volume  =  4/3*r3. 
Cylinder:     volume  =  T2/. 


Length. 


i  centimeter  (cm.)  =  0.3937  in- 
i   meter   (m.)  =  3.281  ft. 

I  kilometer  (km.)    —  0,6214  mi. 
i  micron  (ut)  =  o.ooi  mm. 

=  0.00394  in 


I  sq.  cm.  =0.1550   sq.   in. 
I   sq.  m.   =  10.674  sq.  ft. 


Area. 


i  inch  (in.)     =     2.540  cm. 
i   foot   (ft.)    =  0.3048  m. 
i  mile   (mi.)  =   1.609   km. 
i   mil  =  o.ooi  in. 

=  0.00254  cm. 


i    sq.    in.  =   6.451    sq.   cm. 
i   sq.  ft.    =  0.09290  sq.  m. 


Volume. 


i  cc.  =  0.06103  cu.  in. 

i  cu.  m.  =  35-317  cu.  ft. 

i  liter  (1000  cc.)  =  1.7608  pints. 


i  cu.  in. 
i  cu.  ft. 
i  quart 


16.386    cc. 
0.02832  cu.   m. 
1.1359   liters. 


Mass. 


i    gram    (gm.)         :    15.43    gr. 
i    kilogram    (kg.)  =    2.2046    Ib. 


i    grain    (gr.)    =    0.06480    gm. 
i    pound    (lb.)=  0.45359  kg. 


Density. 

i    gm.   per   cc.  =  62.425  Ib.  per  cu.  ft. 
i  Ib,  per  cu.  ft.   ==   0.01602   gm.   per  cc. 

Force. 

i   gram's   weight    (gm.  wt.)    =   980.6  dynes    (go  =  980.6  cm./sec.2.) 
i  pound's  weight  (Ib.  wt.)       =  0.4448  megadynes  (go  =  980.6.) 

(The  "gm.  wt.'  is  here  denned  as  the  force  of  gravity  acting  on 
a  gram  of  matter  at  sea-level  and  45°  latitude.  The  "Ib.  wt."  is 
similarly  defined.) 


MEASUREMENT  OF  PHYSICAL  QUANTITIES. 

TABLE  II. 


131 


USEFUL  NUMERICAL  RELATIONS. 
Pressure  and  Stress. 


i  cm.  of  mercury  at  o°C. 

=  J3-596  gm.  wt.  per  sq.  cm. 
=  0.19338  Ib.  wt.  per  sq.  in. 


i  in.  of  mercury  at  o°C. 

=  34-533  gm.  wt.  per  sq.  cm. 
=  0.49118  Ib.  wt.  per  sq.  in. 


Work  and  Energy. 

kilogram-meter  (kg.  m.)  =  7.233  ft.  Ib. 

foot-pound  (ft.  Ib.)  =  0.13826  kg.  m. 

joule  =   io7  ergs. 

foot-pound  =  1-3557  X  io7  ergs.  (go  =  980.6  cm./sec.2.) 

foot-pound  =  1-3557  joules  (go  =  980.6.) 

joule  =  0.7376  ft.  Ib  (go  =  980.6.) 


Power  (or  Activity). 

I  horse-power   (H.   P.)  =   3  3000  ft.  Ib.  per  min. 

i  watt   =    i   joule   per  sec.  =   IOT  ergs  per  sec. 

i  horse-power  =   745.64  watts    (go  =  980.6  cm./sec.2) 

i  wa,tt  =  44.28  ft.  Ib.  per  min.  (g0  =  980.6) 


Thermometric  Scales. 


C  =  5/9(F-32)  | 

(C     =     centigrade    temperature; 


F   =   9/5C    +    32. 
F  =  Fahrenheit  temperature.) 


Mechanical  Equivalent. 

I    gm. -calorie    =   4.187    X    io7   ergs. 

=  0.4269  kg.  m.  (g0  =  980.6  cm./sec.2) 
=  3-088  ft.  Ib.   (g0  =  980.6.) 


132  MEASUREMENT  OF  PHYSICAL  QUANTITIES. 

TABLE  III. 
DENSITY  OF  DRY  AIR. 

(Values  are  given  in  gms.  per  cc.) 


Temp. 

Barometric  Pressure  (Centimeters  of  Mercury) 

C. 

72 

73 

74 

75 

76 

77 

0° 

.001225 

.001242 

.001259 

.001276 

.001293 

.001310 

i 

220 

237 

254 

271 

288 

305 

2 

216 

233 

250 

267 

283 

300 

3 

212 

228 

245 

262 

279 

296 

4 

207 

224 

241 

257 

274 

290 

5° 

.OOI2O3 

.001219 

.001236 

.001253 

.001270 

.001286 

6 

198 

215 

232 

248 

265 

282 

7 

194 

211 

227 

244 

260 

277 

8 

190 

206 

223 

239 

256 

272 

9 

186 

202 

219 

235 

251 

268 

10° 

.001181 

.001198 

.001214 

.001231 

.001247 

.001263 

ii 

177 

194 

210 

226 

243 

259 

12 

173 

l89 

2O6 

222 

238 

255 

13 

169 

202 

218 

234 

250 

14 

165 

181 

197 

214 

230 

246 

15° 

.001161 

.001177 

.001193 

.001209 

.001225 

.001242 

16 

i57 

173 

I89 

205 

221 

237 

17 

169 

185 

201 

217 

233 

18 

149 

165 

181 

197 

213 

229 

19 

145 

161 

177 

I 

93 

209 

224 

20° 

.001141 

.001157 

.001173 

.0011 

89 

.001204 

.001220 

21 

137 

i53 

169 

185 

200 

216 

22 

133 

149 

165 

I 

& 

I96 

212 

23 

130 

145 

161 

177 

208 

24 

126 

141 

157 

173 

~~^88 

204 

25° 

.001122 

.001138 

.001153 

.001169 

.001184 

.001200 

26 

118 

i34 

149 

165 

1  80 

196 

27 

114 

130 

145 

161 

176 

192 

28 

I  IO 

126 

142 

157 

172 

188 

29 

107 

122 

138 

153 

169 

184 

30° 

.001103 

.001119 

.001134 

.001149 

.001165 

.001180 

Corrections  for  Moisture  in  the  Atmosphere. 

Dew-point 

Subtract 

Dew-point.  Subtract. 

Dew-point. 

Subtract. 

—  10° 

.00000  i 

+2°     .000003 

+  14° 

.  000007 

g 

2 

+  4          4 

+  16 

8 

—  6 

2 

+  6          4 

+  18 

9 

-  4 

2 

+  8          5 

+  20 

.000010 

2 

3 

+  10         6 

+24 

13 

0 

3 

+  12             6 

+28 

16 

MEASUREMENT  OF  PHYSICAL  QUANTITIES. 


133 


TABLE  IV. 


DENSITIES  AND  THERMAL  PROPERTIES  OF  GASES. 

The  densities  are  given  at  o°C.  and  76  cm.  pressure,  and  the 
specific  heats  at  ordinary  temperatures.  The  coefficients  of  cubical 
expansion  (at  constant  pressure)  of  the  gases  listed  below  are  not 
given  in  this  Table;  they  are  about  the  same  for  all  the  permanent 
gases,  being  approximately  1/273  or  0.003663,  if  referred  in  each 
case  to  the  volume  of  the  gas  at  o°C. 


Gas  or  Vapor. 

Formula 

Density 

(gms.  per  cc.) 

Molecular 
Weight 

Cp-Cv 

CP 

(cals. 
pergm.) 

Air 

_ 

0.001293 



1.41 

0.237 

Ammonia 

NH3 

O.OOO77O 

17.06 

•33 

•530 

Carbon  dioxide 

CO, 

O.OOI974 

44.00 

.29 

.203 

Carbon  monoxide 

CO 

O.OOI234 

28.00 

.40 

•243 

Chlorine 

CU 

0.003133 

70.90 

•32 

.124 

Hydrochloric  acid 

HC1 

0.001616 

36.46 

•40 

.194 

Hydrogen 

H2 

0.0000896 

2.016 

•41 

3410 

Hydrogen    sulphide 
Nitrogen,  pure 

H2S 

N, 

0.001476 
0.001254 

34-08 
28.08 

•34 
.41 

•^245 
.244 

Nitrogen,  atmospheric 

— 

0.001257 



Oxygen 

0, 

0.001430 

32.00 

.41 

.218 

Steam  (ioo°C.) 

H2O 

0.000581 

18.02 

.28 

.421 

Sulphur  dioxide 

SO, 

0.002785 

64.06 

.26 

•154 

TABLE  V. 


DENSITY    AND    SPECIFIC    VOLUME    OF    WATER. 


Temp. 
C. 

Density 

(gms.  per  cc.) 

Specific 
Volume 

(cc.  per  gm.) 

Temp. 
C. 

Density 

(gms.  per  cc.) 

Specific 
Volume 

(cc.  per  gm.) 

0° 

0.999868 

I.OOOI32-' 

20° 

0.99823 

I.OOI77 

i 

927 

073 

25 

777 

294 

2 

968 

032 

30 

567 

435 

3 

992 

008 

35 

406 

598 

3.98 

I.OOOOOO 

f.        000 

40 

224 

782 

5 

.999992 

008 

50 

.98807 

1.01207 

6 

968 

032 

60 

324 

705 

7 

929 

071   / 

70 

.97781 

1.02270 

8 

876 

124.  rf 

80 

183 

902 

9 

808 

192 

90 

.96534 

1.03590 

10 

727 

273 

IOO 

.95838 

1-04343 

15 

126 

874 

102 

693 

5oi 

134 


MEASUREMENT  OP  PHYSICAL  .QUANTITIES. 


TABLE  VI. 


DENSITIES   AND   THERMAL  PROPERTIES   OF   LIQUIDS. 

The  values  given  in  this  Table  are  mostly  for  pure  specimens 
of  the  liquids  listed.  The  student  should  not  expect  the  properties 
of  the  average  laboratory  specimen  to  correspond  exactly  in  value 
with  them.  With  a  few  exceptions  the  densities  are  given  for  ordi- 
nary atmospheric  temperature  and  pressure.  The  specific  heats  and 
coefficients  of  expansion  are  in  most  cases  the  average  values 
between  o°  and  ioo°C.  The  boiling  points  are  given  for  atmos- 
pheric pressure,  and.  the  heats  of  vaporization  are  given  at  these 
boiling  points. 


d 

w 

w       o 

hjW 

<3 

CD 

n* 

£n    8 

o€2. 

^  f& 

T3   P 

C« 

rl-   ~° 

P  g.    3j 

rt-   5' 

O   •~1' 

Liquid 

* 

O 

C/)    O    *~*"1  *^' 

CKJ 

ra 

^3                   r^- 

H*'- 

O 

calories 

3 

pergm. 

degrees 

cals. 

gms.  per  cc. 

per  deg. 

per  degree  C. 

C. 

per  gm. 

Alcohol    (ethyl) 

0.794 

.68 

.00111 

78 

205* 

(methyl) 

•796 

.60 

.00143 

66 

262f 

Benzene 

.880 

.42 

.00123 

80 

93-2 

Carbon  bisulphide 

1.29 

.24 

.OOI2O 

46.6 

84 

Ether 
Glycerine 

•74   (o°) 
1.26 

.00162 
.000534 

35 

90 

Hydrochloric  acid 

1.27 

•75 

.000455 

no 

Mercury 

13.596  (o°) 

•033, 

.OOOl8l5 

357 

67 

Olive  oil 

.918 

.0007,21 

Nitric  acid 

1.56 

:66 

.00125 

86 

US 

Sea-water 

1.025 

.938 

Sulphuric  acid 

•33 

.00056 

338 

122 

Turpentine 
Water 

See    Tab.V. 

•47 

I.OO 

.OOIO5 

See  Tab.  V. 

159 

TOO 

70 

537 

*  The  heat- of  vaporization  of  ethyl  alcohol  at  o°C.  is  236.5. 
t  The  heat  of  vaporization  of  methyl  alcohol  at  o°C.  is  289.2. 


"MEASUREMENT  OF  PHYSICAL-  QUANTITIES. 


-135 


TABLE  VII. 


DENSITIES  AND  THERMAL  PROPERTIES  OF  SOLIDS. 

The  values  given  in  this  Table  are  mostly  for  pure  specimens 
of  the  substances  listed.  The  student  should  not  expect  the  prop- 
erties of  the  average  laboratory  specimen  to  correspond  exactly 
in  value  with  them.  As  a  rule  the  densities  are  given  for  ordi- 
nary atmospheric  temperature  and  pressure.  The  specific  heats 
and  coefficients  of  expansion  are  in  most  cases  the  average  values 
between  o°  and  ioo°C.  The  melting  points  and  heats  of  fusion  are 
given  for  atmospheric  pressure. 


8 

01 

<T>  ™ 
p  11. 

Ir  ? 

^3      2L 

2.*"*  3 

£.  "-K  rD 

Solid. 

*•<" 

I"1"  3"> 
O 

r|'| 

O         !IL. 
3 

3              3 

cals.  per 

«••• 

cals.  per 

gms  per  cc. 

gm. 

per  degree  C. 

degrees  C 

gm. 

Aluminum 

2.70 

O.2I9 

.0000231 

658 

Brass,  cast 

8.44 

.092 

.0000188 

,  drawn 

8.2P 

.092 

.OOOOI93 

Copper 
German-silver 

8.92 
8.62 

•094 

.0946 

.0000172 
.OOOOlS 

IO90 
860 

43-0 

Glass,    crown 

2.6 

.OOOOC90- 

"     ,  flint 

3-9 

.117 

.0000079 

Gold 
Hyposul.   of  soda 

19-3 
1,71 

.0316 

•445 

.0000144 

1065 
48 

Ice 

.918 

•502      . 

.000051 

0 

.80. 

Iron,    cast 

7-4 

•113 

.0000106 

1  100 

23-33 

,  wrought 

7.8 

•US 

.000012 

1600 

Lead 

•0315 

.000029 

326 

5-4 

Mercury 

.0319 

—39 

2.8 

Nickel 

8.90 

.109 

.OOOOI28 

1480 

4-6 

Paraffin,  wax 

.90 

.560 

.OOOOO8-23 

52 

35-1 

,  liquid 

.710 

Platinum 

21.50 

.0324 

.OOOOC90 

1760 

27.2 

Rubber,   hard 

1.22 

•331 

.000064 

Silver 

10-53 

.056 

.0000193 

960 

2'  1.  1 

Sodium    chloride 

2.17 

.214 

.000040 

"800 

Steel 

7-8 

.118 

.OOOO  1  1 

1375 

Wood's    alloy,    solid 

9.78 

•0352 

75-5 

8.40 

"     ,  liquid 

.0426 

1 

,,o1> 


• 


136 


MEASUREMENT  OF  PHYSICAL  QUANTITIES. 


TABLE  VIII . 


SURFACE    TENSION    OF    PURE    WATER    IN    CONTACT 

WITH  AIR. 


Temp. 
C. 

Tension 

(dynes  per  cm.) 

Temp. 
C. 

Tension 
(dynes  per  cm.) 

Temp. 
C. 

Tension 

(dynes  per  cm.) 

0° 

5 

75-5 
74-8 

30° 
35 

71.0 
70.3 

60° 
65 

66.0 
65.1 

10 

74-0 

40 

69.5 

70 

64.2 

15 

73-3 

45 

68.6 

80 

62.3 

20 

72-5 

50 

67.8 

100 

56.0 

«5 

71.8 

55 

66.9 

Crit.  Temp. 

o.o 

TABLE  IX. 


SURFACE  TENSIONS    OF   SOME   LIQUIDS   IN    CONTACT 

WITH  AIR. 


Dynes 
per  cm. 

Dynes 
per  cm. 

Alcohol  (ethyl) 
Alcohol    (methyl) 
Benzene 
Glycerine 

at  20° 
at  20° 
at  15° 
at  18° 

22-24 
22-24 
28-30 
63-65 

Mercury 
Olive  oil 
Petroleum 
Water  (pure) 

at  20° 
at  20° 
at  20° 
at  20° 

470-500 
32-36 
24-26 
72-74 

TABLE  X. 


VISCOSITY  OF  WATER. 


Temp. 
C. 

Coeff.  of  Vise. 

(C.G.S.  Units) 

Temp. 
C. 

Coeff.  of  Vise. 

(C.G.S.  Units) 

Temp. 
C. 

Coeff.  of  Vise. 

(C.  G.S.  Units) 

0° 

5 

10 

15 

20 

O.OI78 
.0151 
.0131 
.0113 
.0100 

25° 

30 

35 
40 

50 

0.0089 
.0080 

.0072 

.0066 

•0055 

60° 
70 
80 
90 

100 

O.OO47 
.0041 
.0036 
.0032 
.0028 

MEASUREMENT  OF  PHYSICAL  QUANTITIES. 

TABLE  XL 
VISCOSITY  OF  AQUEOUS  SOLUTIONS  OF  SUGAR. 


137 


%  Sugar 

Coeff.  at  20°C. 
(C.  G.  S.  Units.) 

Coeff.  at  3o°C. 
(C.  G.  S.  Units.) 

0 

5 

10 
20 
40 

O.OIOO 

.0117 
.0132 
.0191 
.0600 

0.0080 
.0089 
.0104 
.0145 

.0423 

TABLE  XII. 
COEFFICIENTS  OF  FRICTION. 


Substances. 

Static  Coefficient. 

Kinetic    Coefficient. 

Metals  on  metals    (dry) 

from  0.2  to    0.4 

from  0.18  to  0.35 

(wet) 

"     0.15 

0-3 

' 

0.14 

'    0.28 

"       (oiled) 

0.15 

0.2 

' 

0.14 

'  0.18 

Wood  on  wood  (dry) 

(a)     direction  of  fiber 

"     0.5 

0.7 

< 

0.2 

'  0-3 

(b)     normal    to    fiber 

0.4 

0.6 

1 

0.18 

'  0-3 

Leather  belt  on  wood  pulley 

0.45 

0.6 

' 

0.3 

'  0.5 

"     "       iron       " 

"     0.25 

0-35 

0.2 

'  0.3 

TABLE  XIII. 

ELASTIC    CONSTANTS    OF   SOLIDS. 
(Approximate  Values.) 


Substance. 

Bulk-Modulus. 
(C.  G,  S.  Units.) 

Simple  Rigidity. 
(C.  G.  S.  Units.) 

Young's  Modulus. 
(C.  G.  S.  Units.) 

Aluminum 
Brass,    drawn 
Copper 

5-5  x  ic" 
10.8  x     " 
16.8  x     " 

2.5  x  IOU 
37  x 

4-5  x 

6.5  x  10  u 
10.8  x 
12.3  x 

T0  o             t 

fMocc 

•5  x      ( 

Iron,  wrought 
Steel 

14.6  x     " 

18.4  x 

2.4  x 
77  x 

8.2    X 

7.0  x 
19.6  x 
21.4  x 

138 


MEASUREMENT  OF  PHYSICAL  QUANTITIES. 


TABLE  XIV. 

(a)  BOILING  POINT  OF  WATER  AT  DIFFERENT  BARO- 

METRIC   PRESSURES. 

(b)  VAPOR-TENSION  OF  SATURATED  WATER- VAPOR. 

(This  table  may  be  used  either  (a)  to  find  the  boiling  point  t  of 
water  under  the  barometric  pressure  P,  or  (b)  to  .find  the  vapor-tension 
P  of  water-vapor  saturated  at  the  temperature  t,  the  dew-point.) 


t° 

Cent. 

P 

cm. 

D 

gm.|cc. 

t° 
Cent. 

P 

cm. 

D 

gm.|cc. 

t° 

Cent. 

P 

cm. 

D 

gm.|cc. 

—  10° 

.22 

2.3XIO'6 

30° 

3-15 

30.ixio~6 

88?  5 

49.62 

—  9 

23 

2.5x    1 

35 

4.18 

39-3X  " 

89 

50.58 

—  8 

•25 

40 

5-49 

50-9X  " 

89-5 

51-55 

—  7 

•27 

2.9X     ' 

7.14 

65-3x  " 

90 

52-54 

428.4xio~" 

—  6 

.29 

3-2X      ' 

50 

9.20 

83-Ox  " 

90-5 

53-55 

—  5 

•32 

3-4x    ' 

55 

1  1  *y$ 

I04.6x  " 

91 

54-57 

—  4 

•34 

3-7x  " 

60 

14.88 

I3o.7x  " 

91-5 

55-6i 

—  3 

-37 

4.ox  " 

65 

18.70 

I62.IX    " 

92 

56.67 

2 

•39 

4.2X    " 

70 

23-31 

I99-5X  " 

92-5 

57-74 

I 

.42 

4-5x    ' 

24.36 

93 

58-83 

O 

46 

4-9x    ' 

72 

25-43 

93-5 

59.96 

I 

•49 

5-2x    ' 

73 

26.54 

94 

6  1.  06 

2 

-53 

5-6x    ' 

74 

27.69 

94-5 

62.20 

3 

•57 

6.ox    ' 

75 

28.88 

243-7     " 

95 

63-36 

511.1 

4 

.61 

6.4x    ' 

75-5 

29.49 

95-5 

64.54 

5 

.65 

6.8x    ' 

76 

30.11 

96 

6574 

6 

.70 

7-3x    ' 

76.5 

30-74 

96.5 

66.95 

7 

-75 

7-7x    ] 

77 

31.38 

97 

68.18 

8 

.8q 

77-5 

32.04 

97-5 

69.42 

Wo 

•85. 
,.91 

87x    ' 
9-3x    ' 

78 

78-5 

32.71 
33-38 

98 
98.2 

70.71 
71.23 

n 

.98 

IO.QX    ' 

79 

34-07 

98.4 

71-74 

12 

.04 

io.6x    ' 

79-5 

34-77 

98.6 

72.26 

13 

.n 

II.  2X     ' 

80 

35-49 

295-9    " 

98.8 

72-79 

14 

.19 

12.  OX    " 

80.5 

36.21 

99 

73-32 

15 

•2? 

I2.8X    " 

81 

36.95 

99-2 

73-85 

16 

-35 

13.  5x  " 

81.5 

37-70 

99-4 

74.38 

17 

•44 

I4-4X  " 

82 

38.46 

99-6 

74.92 

18 

-53 

I5.2x  " 

82.5 

39-24 

99-8 

75-47 

19 

-63 

i6.2x  " 

83 

40.03 

IOO 

76.00 

606.2   " 

20 

*74 

17.  2X    " 

83.5 

40.83 

100.2 

76.55 

21 

-85 

I8.2X     ' 

84 

41.65 

100.4 

77.10 

22 

-96 

I9-3X    ' 

84-5 

42.47 

100.6 

77.65 

23 

2.09 

20.4X     ' 

85 

43-32 

357-1     " 

100.8 

78.21 

24 

2.22 

21.  6x    ' 

85-5 

44.13* 

101 

78.77 

25 

2-35 

22.9X     ' 

86 

45-05 

102 

8  1.  60 

26 

2-50 

24.2X     ' 

86.5 

45-93 

103 

84.53 

27 

2.65 

25.  6x    ' 

87 

46-83 

105 

90.64 

715.4   " 

28 

2.8l 

27.  ox    ' 

87-5 

47-74 

107 

97.11 

29 

2-97 

28.5X    ' 

88 

48.68 

no 

107-54 

840.1    " 

MEASUREMENT  OF  PHYSICAL  QUANTITIES. 


139 


TABLE  XV. 
THE  WET-  AND  DRY-BULB  HYGROMETER.  DEW-POINT. 

This  Table  gives  the  vapor-pressure,  in  mercurial  centimeters,  of 
the  water-vapor  in  the  atmosphere  corresponding  to  the  dry-bulb 
reading  t°C.  (first  column)  and  the  difference  (first  row)  between 
the  dry-bulb  and  wet-bulb  readings  of  the  hygrometer.  Having 
obtained  from  this  Table  the  value  of  the  vapor-pressure  for  a  given 
case,  the  dew-point  can  be  found  by  consulting  Table  XIV.  The  data 
given  below  are  calculated  for  a  barometric  pressure  equal  to  76  cm. 


t°c. 

Difference  between   Dry-bulb  and   Wet-bulb   Readings. 

0°- 

1° 

2° 

S°- 

4° 

6° 

6° 

7°- 

8° 

9° 

10* 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

cm. 

10° 

•92 

.81 

.70 

.60 

•50 

40 

•31 

.22 

•13 

ii 

.98 

87 

76 

65 

7C 

.26 

17 

^ 

12 

.v^w 

•05 

•  *~>/ 

•93 

*/  w 

.82 

•V-'O 

.71 

!6o 

•50 

•oo 

.40 

•30 

•  r  / 
.21 

.12 

•P3 

13 

.12 

I.OO 

.89 

.76 

•65 

•55 

•45 

•35 

•25 

.16 

.07 

14 

.19 

1.07 

•94 

.83 

•7i 

.61 

•50 

.40 

•30 

.20 

.11 

15 

.27 

1.14 

I.OI 

.90 

•78 

.66 

•55 

•45 

•34 

•25 

•  15 

16 

•35 

1.22 

1.09 

•97 

•84 

•73 

.60 

•50 

.40 

•30 

.19 

i? 

•44 

1-30 

1.  17 

1.04 

.91 

.80 

•67 

•56 

•45 

•35 

•24 

18 

•54 

i-39 

1-25 

1.  12 

•99 

.86 

•74 

•63 

•51 

.40 

•30 

ig 

•63 

1.49 

i-34 

1.20 

1.07 

•94 

.81 

•69 

•57 

46 

•35 

20 

•74 

1-59 

i-43 

1.29 

i.  02 

.88 

•76  . 

•64 

•52 

.41 

21 

•85 

1.69 

1-53 

1.38 

1.24 

1.  10 

•96 

•84 

•71 

•47 

22 

1.97 

i.  80 

1.64 

1.48 

1-33 

1.19 

•05 

?9iJ 

.66 

•54 

23 

2.09 

1.92 

1-75 

1-59 

1-43 

1.28 

•13 

I.OO 

.86 

•73 

.61 

24 

2.22 

2.04 

1.86 

1.70 

i.  .53 

1.38 

•23 

1.09 

•94 

.81 

.68 

25 

2-35 

2.17 

1.99 

1.81 

1.64 

1.48 

•33 

1.18 

1.03 

.90 

-76 

26 

2.50 

2.31 

2.  II 

1.94 

1.76 

1-59 

•43 

1.28 

1.13 

•98 

•84 

27 

2.65 

2-45 

2.25 

2.07 

1.88 

1.71 

1.38 

1-23 

i.  08 

•93 

28 

2.81 

2.60 

2.40 

2.20 

2.01 

1.83 

i  66 

1.49 

1-33 

1.18 

i.  02 

29 

2.98  ' 

2.76 

2-55 

2-35 

2.15 

1.96 

1.78 

1.61 

1.44 

1.28 

1.  12 

30 

3-15 

2-93 

2.71 

2.50 

2.29 

2.IO 

1.91 

1-73 

1-55 

1-39 

1.23 

TABLE  XVI. 

Miscellaneous. 

(i).    Heat  of  Neutralization. 

Any  strong  acid  with  any  strong  alkali  evolves  (  +  )  about 

761  calories  for  every  gm.  of  water  formed. 
(2.)     Heat  of  Solution. 

For  Calcium  oxide   (CaO),  +  327  cals.  per  gm. 

"     Sodium  chloride   (NaCl),  —    21      "        "      " 

"       hydroxide    (NaOH),          +  248      "        "      " 
"     hyposulphite  (Na«S»O«  +  5H2O),—    44.8  " 
(3).    Lowering  of  Freezing  Point  of  Water. 

For  dilute  aqueous  solutions  of  any  salt  the  lowering 
is  proportional  to  the  mass  of  salt  dissolved  in  the  same 
mass  of  water.  If  ionization  does  not  occur,  each  gram- 
molecular  weight  of  the  salt  in  TOGO  gm.  of  water  will 
lower  the  freezing  point  1.86°.  If  ionization  occurs,  how- 
ever, the  lowering  is  increased  two,  three,  or  more  times, 
depending  upon  the  number  of  separate  parts  or  ions  into 
which  the  molecule  of  the  salt  is  divided  by  the  water. 


I4O  MEASUREMENT  OF   PHYSICAL  QUANTITIES. 

TABLE  XVII. 
Natural   Sines. 


0' 

6' 

12 

18 

24 

30' 

36 

42 

48' 

54' 

123 

4       5 

0° 

0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

oi57 

369 

12      15 

1 

2 
3 

0175 
0349 
0523 

0192 
0366 
0541 

0209  0227 
0384^0401 

05580576 

0244 
0419 
0593 

0262 
0436 
0610 

0279 

0454 
0628 

029703140332 
0471  04880506 
0645  0663  O68O 

369 
369 
369 

12      15 

12      15 
12-      15 

4 
5 
6 

0698 
0872 
1045 

0715107320750 
0889  0906  0924 
1063!  io8o|  1097 

0767  0785 
0941  0958 
1115  1132 

0802 
0976 
"49 

0819 

0993 
1167 

0837 
ion 
1184 

0854 
1028 

I2OI 

369 
369 
369 

12      15 
12      14 
12      14 

7 
8 
9 

1219 
1392 
1564 
1736 

1236 
1409 
1582 

1253 
1426 

1599 

1271 
1444 
1616 

1288 
1461 
1633 

1305 
1478 
1650 

1323 
1495 
1668 

1340 
1513 
1685 

1357 
1530 
1702 

1374 
1547 
1719 

369 
369 
369 

12      14 
12      14 
12      14 

10 

1754 

1771 

1788 

1805 

1822 

1840 

i857|i874 

1891 

369 

12      14 

11 
12 
13 

1908 
2079 

2250 

1925 
2096 
2267 
2436 
2605 
2773 

I942JI959 
2113  2130 
2284  2300 

1977 
2147 
2317 

1994 
2164 
2334 

2OII 

2181 

2351 

2028 
2198 
2368 

2045 

2215 

2385 

2O62 
2232 
2402 

369 
369 
368 

II      14 
II       14 
II       14 

14 
15 
16 

2419 

2588 
2756 

2453 
2622 
2790 

2470 
2639 
2807 

2487 
2656 
2823 

2504 
2672 
2840 

2521 

2689 
2857 

2538 
2706 
2874 

2554 
2723 
2890 

2571 
2740 
2907 

368 
368 
3  6  8 

II       14 
II       14 
II       14 

17 
18 
19 

2924 

3090 

3256 

2940 
3107 
3272 

3437 

2957 
3123 
3289 

3453 

2974 
3MO 
3305 

2990 
3156 
3322 

3486 

3007 
3173 
3338 

3024 
3190 

3355 

3040 
3206 
337i 

3057 
3223 
3387 

3074 
3239 
3404 

368 
368 

3  5  8 

II       14 

II       14 
II       14 

20 

3420 

3469 

3502 

35i8 

3535 

3551 

3567 

3  5  8 

II       14 

21 
22 
23 

3584 
3746 

39°7 

3600 
3762 
3923 

3616 

3778 
3939 

3633 
3795 
3955 

3649 
3811 
397i 

3665 
3827 
3987 

3681 

3843 
4003 

3697 
3859 
4019 

3714 

3875 
4035 

3730 
3891 
4051 

3  5  8 
3  5  8 
3  5  8 

II       14 
II       14 
II       14 

24 
25 
26 

4067 
4226 
4384 

4083 
4242 
4399 

4099 
4258 
4415 

4H5 
4274 
443i 

4131 
4289 
4446 

4147 
4305 
4462 

4163 
4321 
4478 

4179 
4337 
4493 

4195 
4352 
4509 

42IO 

4368 
4524 

3  5  8 
3  5  8 
3  5  8 

II       13 
II       13 

10    13 

27 
28 
29 

4540 
4695 

4848 

4555 
4710 
4863 

4571 
4726 
4879 

4586 
474i 
4894 

4602 
4756 
4909 

4617 
4772 
4924 

4633 
4787 
4939 

4648 
4802 
4955 

4664 

4818 
4970 

4679 
4833 
4985 

3  5  8 
3  5  8 
3  5  8 

10    13 
10    13 
10    13 

30 

5000 
5150 
5299 
5446 

5015 

5030 

5045 

5060 

5075 

5090 

5105 

5120 

5135 

3  5  8 

10    13 

31 
32 
33 

5165 
53M 
546i 

5180 
5329 
5476 

5195 
5344 
5490 

5210 
5358 
5505 

5225 
5373 
5519 

5240 

5388 
5534 

5255 
5402 

5548 

5270 
5417 
5563 

5284 
5432 

5577 

2  5  7 
2  5  7 
2  5  7 

IO      12 
IO      12 
10      12 

34 
35 
36 

5592 
5736 
5878 

5606 
5750 
5892 

5621 
5764 
5906 

5635 
5779 
5920 

5650 
5793 
5934 

5664 
5807 
5948 

5678 
5821 
5962 

5693  5707  572i 
5835  5850  5864 
5976159906004 

2  5  7 
2  5  7 
2  5  7 

IO      12 
10      12 

9     12 

37 
38 
39 

6018 

6157 
6293 

6428 

6032 
6170 
6307 
6441 

6046 
6184 

6320 

6060 
6198 
6334 

6074 
6211 

6347 

6088 
6225 
6361 
6494 

6101 
6239 
6374 

6ii5l6i296i43 
6252  6266  6280 
6388  6401  6414 

2  5  7 
2  5  7 
247 

9     12 

9     ii 
9     u 

40 

6455 

6468 

6481 

6508 

6521  6534  6547 

247 

9     ii 

41 

42 
43 

6561 
6691 
6820 

65746587 
67046717 

68336845 

6600 
6730 

6858 

6613 

6743 
6871 

6626  6639 
67566769 
688416896 

6652  6665  6678 
6782  6794  6807 
69096921  6934 

247 
246 
246 

9     ii 
9     n 

8     u 

44 

6947 

6959 

6972 

6984 

6997 

7009 

7022 

7034 

7046 

7059 

246 

8     10 

MEASUREMENT  OF  PHYSICAL  QUANTITIES. 

TABLE  XVII.  (Coned.) 
Natural   Sines. 


141 


0 

7071 

6' 

7083 

12 

7096 

18 

24 

30 

36 

42 

7157 

48 

54 

718: 

123 

4  5 

45° 

7108 

7120 

7133 

7M5 

7169 

2    4    6 

8  10 

46 
47 
48 

7193 
73M 
743i 

7206 
7325 
7443 

7218 
7337 
7455 

7230 

7349 
7466 

7242 
7361 
7478 

7254 
7373 
7490 

726f 
7385 
75oi 

7278 
7396 
7513 

7290 
7408 
7524 

7302 
7420 
7536 

246 
246 
246 

8  10 
8  10 
8  10 

49 
50 
51 

7547 
7660 

777i 

7558 
7672 
7782 

7570758i 
7683  7694 
7793  7804 

7593 
7705 
7815 

7604 
7716 
7826 

7615 

7727 
7837 

7627 

7738 
7848 

7638 
7749 

7859 

7649 
7760 
7869 

246 
246 
2    4     5 

8  9 
7  9 
7  9 

52 
53 
54 

7880 
7986 
Sogo 

7891 
7997 
8100 

7902  7912 
8007!  80  1  8 
8nij8i2i 

7923 
8028 
8131 

7934 
8039 
8141 

7944 
8049 
8151 

7955 
8059 
8161 

7965 
8070 
8171 

7976 
8080 
8181 

245 

2    3     5 
2    3     5 

7  9 
7  9 

7  8 

55 

8192 

8202 

8211  8221 

8231 

8241 

8339 
8434 
8526 

8251 
8348 
8443 
8536 

8261 

8271 

828) 

235 

7  8 

56 
57 
58 

8290 
8387 
8480 

8572 
8660 
8746 
8829 
8910 
8988 

8300 
8396 
8490 

S5Si 
8669 
8755 
8838 
8918 
8996 

83108320 
84068415 
84998508 

8329 
8425 
8517 

8358 
8453 
8545 

8368 
8462 
8554 

8377 
8471 
8563 

235 
2    3     5 
235 

6  8 
6  8 
6  8 

59 
60 
61 

8590 
8678 
8763 
8846 
8926 
9003 

8599 
8686 
8771 

8607 
8695 
8780 

8616 

8704 
8788 

8625 
8712 
8796 

8634 

8721 
8805 

8643 
8729 

8813 

8652 
8738 
8821 

3    4 

3    4 
3     4 

6  7 
6  7 
6  7 

62 

63 
64 

8854 

8934 
901  1 

8862 
8942 
9018 

8870 

8949 
9026 

8878 
8957 
9033 
9107 

8886 
8965 
9041 

8894 

8973 
9048 

8902 
8980 
9056 

3     4 
3    4 
3    4 

5  7 
5  6 
5  6 

65 

9063 

9070 

9078 
9J50 
9219 
9285 

9348 
9409 
9466 

9085 

9092 

9100 

9114 

9121 
9191 
9259 
9323 

9128 

2     4 

5  6 

66 
67 
68 

9135 
9205 
9272 

9*43 
9212 
9278 

9157 
9225 
9291 

9164 
9232 
9298 

9171 
9239 
9304 

9178 
9245 
93U 

9184 
9252 
9317 

919^ 
9265 
9330 

2     3 

2      3 
2      3 

5  6 
4  6 
4  5 

69 
70 
71 

9336 
9397 
9455 

9342 
H03 
9461 

9354 
9415 
9472 

936i 
9421 

)478 

9367 
9426 

9483 

9373 
9432 
9489 

9379 
9438 
9494 

9385 
9444 

9391 
9449 
9505 

2      3 

2     3 

2      3 

4  5 
4  5 
4  5 

9500 

72 
73 
74 

95ii 
9563 
9613 

)Vt> 

9568 
9617 

9521 

9573 
9622 

9527 
9578 
9627 

9532 
9583 
9632 

9537 
9588 
9636 

9542 
9593 
9641 

9686 

954« 
9598 
9646 

9553 
9603 
9650 

9558 
9608 
9655 
9699 

2      3 
2      2 
2      2 

4  4 

3  4 
3  4 

75 

9659 
9703 
9744 
9781 

9664 

9668 

9673 

9677 

9681 

9690 

9694 
9736 
9774 
9810 

I       2 

3  4 

76 
77 
78 

9707 
9748 

9785 

9711 
975i 
9789 

9715 
9755 
9792 

9720 

9759 
9796 

9724 
9763 
9799 

9728 
9767 
9803 

9732 
9770 
9806 

9740 
9778 
9813 

2 
2 
2 

3  3 
3  3 
2  3 

79 
80 
81 

9816 

9848 
9877 

9820 

)35i 
9880 

9823 

9854 
9882 

9826 
9857 
9885 

9829 
)S6o 
)888 

)S33 
9863 
9890 

9836 
9866 
9893 

9839 
9869 

9895 

9842 
9871 
9898 

9845 
9874 
9900 

I              2 
O 
O 

2  3 

2  2 
2  2 

82 
83 
84 

9903 
9925 
9945 

9905 
9928 

9947 

99°7 
9930 
9949 

9910 
9932 
995i 

WI2 

9934 
9952 
9968 

9914 
9936 
9954 

9917 
9938 
9956 

9919 
9940 
9957 

9921 
9942 
9959 

9923 
9943 
9960 

O 
O 
O 

2  2 
I  2 
I  I 

85 

9962 

9963 

9965 

9966 

9969 

9971 

9972 

9973 

9974 

O     O      I 

I  I 

86 
87 
88 

9976 
9986 
9994 

)977 
9987 
9995 

9978 
9988 
9995 

9979 
9989 
9996 

9980 
9990 
9996 

9981 
9990 
9997 

9982 
999  i 
9997 

9983 
9992 
9997 

9984 

9993 
9998 

9985 
9993 
9998 

0     0      I 
000 
O     O      O 

I  I 
I  I 
O  O 

89 

9998 

9999 

9999 

9999 

9999 

I.OOO 
nearly 

I.OOO 
nearly 

I.OOO 

nearlv 

I.OOO 

nearly 

I.OOO 
nearly 

O     O      O 

O  0 

142 


MEASUREMENT  OF  PHYSICAL  QUANTITIES. 


TABLE  XVIII. 
Natural   Tangents. 


0' 

6 

12' 

18 

24 

30 

36 

42 

48' 

54' 

123 

4      5 

12          I4- 

0° 

.0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

oi57 

369 

1 

2 
3 

.0175 

•0349 
.0524 

0192 
0367 
0542 

0209 
0384 
0559 

0227 
0402 
0577 

0244 
0419 
0594 

0262 

0437 
0612 

0279 

0454 
0629 

0297 
0472 
0647 

0314 
0489 
0664 

0332 
0507 
0682 

369 
369 
369 

12         I5 
12         I5 
12          15 

4 
5 
6 

.0699 
.0875 
.1051 

0717 
0892 
1069 

0734 
0910 
1086 

0752 
0928 
1104 

0769 
0945 

1122 

0787 
0963 
"39 

0805 
0981 
"57 

0822 
0998 
"75 

0840 
1016 
1192 

0857 
1033 

I2IO 

369 
369 
369 

12         I5 
12         I5 
12          I5 

7 
8 
9 

.1228 
.1405 
.1584 

1246 
1423 
1602 

1263 
1441 
1620 

1281 

1459 
1638 

1299 
1477 
1655 

1317 

1495 
1673 

1334 
1512 
1691 

1352 
1530 
1709 

1370 

1548 
1727 
1908 

1388 
1566 
1745 
1926 

369 
369 
369 

12          I5 
12         J5 
12          I5 

10 

.1763 

1781 

1799 

1817 

1835 

1853 

1871 

1890 

369 

12          I5 

11 
12 
13 

.1944 
.2126 
.2309 

1962 
2144 
2327 

1980 
2162 

2345 

1998 

2180 
2364 

2016 
2199 

2382 

2035 
2217 
2401 

2053 
2235 
2419 

2071 
2254 
2438 

2089 
2272 
2456 

2107 
2290 

247S 

3      6     9 
369 
369 

12          15 
12         15 
12         15 

14 
15 
16 

•2493 
.2679 
.2867 

2512 
2698 
2886 

2530 
2717 
2905 

2549 
2736 
2924 

2568 
2754 
2943 

2586 

2773 
2962 

2605 
2792 
2981 

2623 
28ll 
3000 

2642 
2830 
3019 
3211 
3404 
3600 

2661 
2849 
3p_38 
3230 
3424 
3620 

369 
369 
369 

12         l6 
13         l6 

13       16 

17 
18 
19 

•3057 
•3249 
•3443 

3076 
3269 
3463 

3096 
3288 
3482 

3"5 
3307 
3502 

3134 
3327 
3522 

3153 
3346 
354i 

3172 

3365 
356i 

3I91 

3385 
3581 

3      6    10 
3      6    10 
3610 

13      .16 
13         l6 
13         17 

20 

.3640 

3659 

3679  3699 

3719 

3739 

3759 

3779 

3799 

3819 

3      7    10 

I  3         17 

21 
22 
23 

•3839 
.4040 

•4245 

3859 
4061 
4265 

3879 
4081 
4286 

3899 
4101 

4307 

39J9 
4122 

4327 

3939 
4142 

4348 

3959 
4163 

4369 

3978 
4183 
4390 

4000 
4204 
44" 

4020 
4224 
4431 
4642 
4856 
5073 

3      7    I0 

3      7    10 

3      7    10 

13         17 

>4       17 
14       17 

24 
25 
26 

•4452 
.4663 
.4877 

4473 
4684 
4899 

4494 
4706 
4921 

4515 
4727 
4942 

4536 
4748 
4964 

4557 
4770 
4986 

4578 
479' 
5008 

45994621 

48134834 
50295051 

7    10 
7    ii 
7    " 

14      18 
14      18 
15       18 

27 
28 
29 

•5095 
•5317 
•5543 

5U7 
5340 
5566 

5139 
5362 

5589 

5161 

5384 
5612 

5184 
5407 
5635 

5206 
5430 
5658 

5228 
5452 
5681 

52505272 
5475  5498 
5704  5727 

5295 
5520 
5750 

7    >i 
8    ii 

8       12 

15       18 
"5       J9 
'5       19 

30 

•5774 

5797 

5820 

5844 

5867 

5890 

59M 

5938 
6176 
6420 
6669 

596i 

5985 

4      8    12 

16       20 

31 
32 
33 

.6009 
.6249 
.6494 

6032 
6273 
6519 

60566080 
6297  6322 
6544  6569 

6104 
6346 
6594 

6128 
6371 
6619 

6152 
6395 
6644 

6200 

6445 

6694 

6224 
6469 
6720 

4      8    12 
4      8    12 
4      S    13 

16       20 

l6         20 
17          21 

34 
35 
36 

6745 
.7002 
.7265 

6771 
7028 
7292 

6796 
7054 
7319 

6822 
7080 
7346 

6847 
7107 
7373 

6873 
7133 
7400 

6899 
7159 
7427 

6924 
7186 

7454 

6950 
7212 
748i 

6976 

7239 
7508 

4      9    '3 
4      9    '3 
5      9    H 

17          21 

l8          22 
l8         23 

37 
38 
39 

•7536 
•7813 
.8098 

7563 

7841 
8127 

7590 
7869 
8156 

7618 

7898 
8185 

7646 
7926 
8214 

7673 
7954 
8243 

7701 

7983 
8273 

7729 
8012 
8302 

7757 
8040 
8332 

7785 
8069 
8361 
8662 

5      9    '4 
5    1°    14 
5    i°    i5 

l8         23 
I9         24 
20         24 

40 

.8391 

8421 

8451 

8481 

8511 

8541 

857i 

S6oi 
8910 
9228 
9556 

8632 

5    I0    J5 

20         25 

41 
42 
43 

.8693 
.9004 
•9325 

8724 
9036 
9358 

87548785 
9067  9099 
9391  9424 

8816 
9131 
9457 

8847 
9163 
9490 

8878 
9195 
9523 

8941 
9260 
959<> 

8972 

9293 
9623 

5    10    16 
5    ii    16 
6    ii    17 

21           26 

xi       27 

22         28 

44 

9^57 

9691 

9725 

9759 

9793 

9827 

9861 

9896 

9930 

9965 

6    ii    17 

23         29 

MEASUREMENT  OF  PHYSICAL  QUANTITIES. 

TABLE  XVIII.     (Confd.) 
Natural   Tangents. 


143 


0 

6' 

12 

18 

24' 

30 

36 

42' 

48 

54 

123 

4    5 

45° 

I.OOOO 

0035 

0070 

0105 

0141 

0176 

0212 

0575 
0951 
1343 

0247 

0283 

0319 

6     12      18 

24    3° 

46 
47 
48 

1.0355 

1.0724 
1.  1  1  06 

0392 
0761 
H45 

0428 
0799 
1184 

0464 
0837 
1224 

0501 

0875 
1263 

0538 
0913 
1303 

0612 
0990 

1383 

0649 
1028 
1423 

0686 
1067 
1463 

6      12      18 

6   .13    19 

7    13    20 

25    3« 
25    32 

26     33 

49 
50 
51 

1.1504 
1.1918 

1-2349 

1544 
1960 

2393 

1585 

2OO2 
2437 

1626 
2045 
2482 

1667 
2088 
2527 

1708 
2131 
2572 

I75C 
2174 
2617 

1792 
2218 
2662 

1833 
2261 
2708 

1875 
2305 
2753 

7    '4    21 

7       '4      22 

8    15    23 

28     34 
29     36 
30     38 

52 
53 
54 

1.2799 
1.3270 
1.3764 

2846 
3319 
3814 

2892 
3367 
3865 

2938 
34i6 
3916 

2985 
3465 
3968 

3032 
35H 
4019 

3079 
3564 
4071 

3127 
3613 
4124 

3175 
3663 
4176 

3222 

3713 
4229^ 

18    16   23 
8    16    25 
9    17    26 

3'     39 
33    4» 
34     43 

55 

1.4281 

4335 

4388 

4442 

4496 

4550 

4605 

4659 

4715 

4770 

9    18    27 

36     45 

56 
57 
58 
59 
60 
61 

1.4826 

1-5399 
1.6003 

4882 
5458 
6066 

4938 
5517 
6128 

4994 

5577 
6191 

5051 
5637 
6255 

5108 

5697 
6319 

5166 

5757 
6383 

5224 

5818 
6447 

5282 
5880 
6512 

5340 
594i 
6577 

ip    19    29 

10     20     30 
II      21      32 

38    48 
40    50 
43     53 

1.6643 
1.7321 
1.8040 

6709 

7391 
8115 

6775 
7461 
8190 

6842 
7532 
8265 

6909 
7603 
8341 

6977 

7675 

8418 

7045 

7747 
8495 

7H3 

7820 
8572 

7182 

7893 
8650 

725J 
7966 

8728 

"      23     34 
12     24     36 

13    26    38 

45     56 
48     60 
5'     64 

62 
63 
64 

1.8807 
1.9626 
2.0503 

8887 
97" 
0594 

8967 

9797 
0686 

9°47 

9883 
0778 

9128  9210 
9970  6057 
0872  0965 

9292 
0145 
1060 

9375 
0233 

"55 

9458 
6323 
1251 

9542 
0413 
1348 

14  27  41 

IS    29    44 
|i6    31    47 

55     68 
58    73 
63    78 

65 

2.1445 

1543 

1642 

1742 

1842 

1943 

2045 

2148 

2251 

2355 

\'7    34    5» 

68    85 

66 
67 
68 

2.2460 
2-3559 
2-4751 

2566 
3673 
4876 

2673 
3789 
5002 

2781 

39°6 
5129 

2889 
4023 
5257 

2998 
4142 
5386 

3109 
4262 

5517 

3220 
4383 
5649 

3332 
4504 

5782 

3445 
4627 
59I(: 

18    37    55 

1  20     40     60 

22    43    65 

74    92 
79    99 
87  108 

69 
70 
71 

2.6051 

2-7475 
2.9042 

6187 
7625 
9208 

6325 
7776 
9375 

6464 
7929 
9544 

6605 
8083 
97U 

6746 
8239 
9887 

6889 
8397 
0061 

7034 
8556 
0237 

7179 
8716 
0415 

7326 

8878 
0595 

24    47    7' 
26    52    78 
29    58    87 

95  "8 
104  130 
"5  M4 
129  t6i 
144  180 
162  203 

72 
73 
74 

3-0777 
32709 

3-4874 

0961 
2914 
5105 

1146 
3122 
5339 

1334 
3332 
5576 

1524 
3544 
5816 

1716 

3759 
6059 

1910 
3977 
6305 

2106 
4197 
6554 

2305 
4420 
6806 

2506 

464^ 
7062 

]32     64     96 

I  36    72  108 

1  41     82  122 

75 

3.7321 

7583 

7848 

8118 

8391 

8667 

8947 

9232 

9520 

9812 

1  46    94  »39 

i  86  232 

76 
77 
78 

4.0108 

4-33I5 
4.7046 

0408 
3662 
7453 

0713 
4015 
7867 

J022 

4374 

8288 

1335 
4737 
8716 

1653 
5107 
9152 

1976 

5483 
9594 

2303 
5864 
0045 

2635 
6252 
0504 

2972 
664* 
097C 

I  53  107  160 
J  62  124  186 
)J73  146  219 

214  267 
248  310 
292  365 

79 
80 
81 

5.1446 
5-67I3 
6.313^ 

1929 
7297 
3859 

2422 
7894 
4596 

292^ 
8502 

3435 
9124 
6122 

3955 
9758 
6912 

4486 
0405 

5026 
1066 

8548 

5578 
1742 
9395 

6i4C 
2432 

187175262 

350  437 

5350 

7920 

026^ 

ice  -  col- 
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82 
83 
84 

7-II54 
8.144.. 
9-5I44 

2066 
2636 
9.677 

3002 
3863 
9-845 

3962 
5126 

IO.O2 

4947 
6427 

IO.2O 

5958 
7769 
10.39 

6996 
9152 
10.58 

8062 

0579 
10.78 

9!58 
2052 
10.99 

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85 

11-43 

11.66 

11.91 

I2.I6 

1243 

12.71 

13.00 

13-30 

i3-9f 

86 
87 
88 

14.30 
19.08 
28.64 

14.67 
19.74 
30.14 

15-06 
20.45 
31.82 

15.46 
21.20 
33.69 

15.89 
22.02 

35-80 

16.35 
22.90 
38-19 

16.83 
23.86 
40.92 

17-34 
24.90 
44.07 

17.89 
26.03 
47-74 

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27.2: 
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89 

57-29 

63.66 

71.62 

81.85 

95-49 

114.6 

143-2 

[91.0 

286,5 

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THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
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LIBRARY,  UNIVERSITY  OF  CALIFORNIA,  DAVIS 

BookSlip-50TO-5,'70(N6725s8)458— -  A-31/5 


U8086 


Minor,  R.S. 

Physical  measurements 
in  the  properties  of 


Call  Number: 


QC220 


1*8086 

Minor,  R.S. 

Physical  measurements 
in  the  properties  of 
matter  and  in  heat. 


QC220 
M5 


LIBRARY 

UNIVERSITY   OF    CALIFORNIA 
DAVIS 


